INIS-mf—8683

FJ.F. van Odenhoven

KINETIC THEORYOF TRANSPORT PROCESSES

IN PARTIALLY IONIZED GASES

KINETIC THEORY OF TRANSPORT PROCESSES

IN PARTIALLY IONIZED GASES

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE

TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE

HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR

MAGNIFICUS, PROF.DR. S.T.M. ACKERMANS, VOOR

EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN

DECANEN IN HET OPENBAAR TE VERDEDIGEN OP

VRIJDAG 18 FEBRUARI 1983 TE 16.00 UUR

DOOR

FERDINAND JOAN FRANCISCUS VAN ODENHOVEN

GEBOREN TE EINDHOVEN

DIT PROEFSCHRIFT IS GOEDGEKEURD

DOOR DE PROMOTOREN

PROF.DR.IR. P.P.J.M. SCHRAM

EN

PROF.DR. M.P.H. WEENINK

Wees niet bang voor het

langzaam voorwaarts gaan,

wees slechts bevreesd

voor het blijven staan

(Chinees gezegde)

Contents

page

I Introduction I

References 5

II Basic equations 6

References 15

III Very weakly ionized gases 16

III-l The electron distribution function 17

III-2 The electron macroscopic equations 23

III-3 Form relaxation of the electron distribution 30

III-4 The inclusion of Coulomb collisions 34

References 37

IV Weakly ionized gases 38

IV-1 Heavy particle results 39

IV-2 Perturbation solution of the electron

distribution function 46

IV-3 The macroscopic electron equations 51

IV-4 The first order isotropic correction 57

IV-5 Electron transport coefficients 61

IV-6 Modifications for a seeded plasma 67

References 69

V Strongly ionized gases 70

V-l Heavy particle results 72

V-2 The electron kinetic equation 77

V-3 The electron macroscopic equations 82

V-4 The nonisotropic part of the

electron distribution 86

References 93

VI Numerical results 94

VI-1 The isotropic correction 95

VI-2 Electron transport coefficients 108

VI-3 Electrical conductivity of

cesium seeded argonplasma 118

References 120

VII Summary and conclusions 121

Appendices

A Expansion of electron-heavy particle collision integrals

A-l Electron-atom collisions 124

A-2 Electron-ion collisions 128

A-3 Moments of the electron-heavy particle

collision integral 130

B Some H-theorems and properties of collision integrals

B-l The zeroth order electron-atom

collision operator 133

B-2 The zeroth order electron-ion

collision integral 134

B-3 H-theorems for the ion distribution function 135

C Harmonic tensors 137

D The Landau collision integral for identical particles

D-l The Landau collision integral 140

D-2 The linearized Landau collision operator

for like particles 143

D-3 Matrix elements for the operators obtained from

the Landau collision integral 147

E Renormalization of the ion multiple collision term 150

References to the appendices 151

Samenvatting .152

Nawoord 154

Korte levensloop 154

-1-

I INTRODUCTION

One can state that the modern kinetic theory of non-equili-

brium processes in dilute gases came to maturity with the works

of Chapman and Enskog^The book by Chapman and Cowling2 has

never ceased to be an Indispensable textbook on this matter.

Since then there have been written many new textbooks , and

much has been added to the theory, especially to the kinetic

theory of plasmas. More complete historical summaries can be

found in the references^.

The method of multiple scales is one of the important tools

used in this thesis. First introduced by Sandri e.a.4 it has

developed into a valuable mathematical device5. It has also

proved to be very succesful in deriving kinetic equations from

the BBGKY-hierarchy6.

The purpose of the present work is the description of transport

processes and the calculation of transport coefficients of

partially ionized gases. The calculations are restricted to

elastic collision processes. This is certainly justified if the

kinetic energy of the electrons is much smaller than the

excitation energy of the first atomic energy level. There are

of course, always inelastic collisions involving high energy

electrons, but their influence on the values of the transport

coefficients is small, because these result from integrations

over the entire velocity space.

In chapter II the basic equations and the multiple time scale

formalism are expounded. The electrons are of special interest,

since they contribute significantly to all transport processes.

Because of their small mass the electrons often have a tempera-

ture different than the one of the heavy particles. If there

are only very few electrons the isotropic part of the electron

distribution function can deviate significantly from an equili-

brium Maxwelllan as a consequence of fields, gradients and

temperature differences which may be present. There are two

limiting cases in which the situation is relatively simple.

-2-

In the fully ionized or Spltzer limit the isotropic part of the

electron distribution function is a Maxwellian and the non-

isotropic part has been computed numerically by Spitzer and

HSrm^. Within the framework of the Landau kinetic equation this

solution is exact.

In the Lorentz limit (very small degree of ionization but

finite electron-atom mass ratio), on the other hand, the

isotropic part is found to be a so-called Davydov distribution

function8. If the neutrals are sufficiently cold, the

Druyvesteyn distribution is a special case of this distribution

for the hard spheres interaction model.

One can distinguish four domains for the electron density with

different orderings in terms of the small parameter e which is

the square root of the electron-atom mass ratio:

hG = (m /m )6 Si

(i-D

Two of these domains contain the already mentioned cases of

very low respectively high degree of ionization. The

definition of the different regions in terms of the ratio of

the electron-electron to electron-atom collision frequency,

which is proportional to the electron-atom density ratio, is

now as follows:

Very WeaklyIonized Gas

Vee „ 2<< e

Vea

NonlinearRegion

" 6 e =#(e 2)ea

Weakly IonizedGas

ea

Strongly IonizedGas

V

ea

Adjacent to the region of the very weakly ionized gases lies a

region where the equation for the electron distribution

function in zeroth order of e is non-linear and the form of the

distribution function varies with the electron density between

a Davydov and a Maxwell distribution.

In chapter III the first two regions are considered. An order-

ing different from the work of van de Water10 is assumed. Some

results additional to his are obtained.

-3-

The strongly Ionized domain is defined as the region where all

collision frequencies of the electrons are of the same order of

magnitude. This region is investigated in chapter V.

It contains as a special case the fully ionized limit, as far

as the electron equations are concerned.

The equation determining the nonisotropic part of the electron

distribution function is written in the form of a differential

equation, which permits easier calculations. In the fully

ionized limit the integro-differential equation solved at first

by Spitzer and Harm is shown to reduce to a simple second order

differential equation.

Between this region and the nonlinear one a fourth region of

interest is situated. Here the electron mutual collision

frequency is smaller than the electron-atom collision frequency

by a factor e. Plasmas in this region are referred to as weakly

ionized. The interesting feature of this region is the appear-

ance of an isotropic correction to the Maxwellian distribution

function which is found in zeroth order of e.

The necessity of an isotropic correction had already been

indicated by van de Water10. His work was, however, restricted

to a Lorentz like plasma with Maxwell interaction between

electrons and atoms. The equation for this isotropic correction

is solved analytically in chapter IV. This correction leads to

contributions to the transport coefficients which are nonlinear

in the fields and gradients. In this way one gets for instance

a correction to the electrical conductivity which depends

quadratically on the electric field. There also appear new

transport processes partly also nonlinearly depending on fields

and gradients. The Onsager symmetry relations do not hold for

these contributions to the transport cofficients. Other contri-

butions are due to the influence of the Coulomb collisions on

the electron-atom collisions, i.e. multiple collisions. These

are linear and obey Onsagers' theorem.

Much work in the field of transport coefficients in partially

fts was motivated by the possibility of direct energy

-4-

conversion by means of an MHD-generator'1. Therefore some

attention is also paid in this thesis to new transport

processes and higher order corrections to transport coeffi-

cients in alkali seeded noble gas plasmas. This attention is

rewarding, because for these plasmas a better comparison with

experiments appears to be possible.

All results of the calculations and the comparisons with

experiments are collected in chapter VI.

The method used in this thesis consists of an expansion of the

unknown quantities into an asymptotic series in the small

parameter e supplemented by the method of multiple time scales.

The general form of the solution f of the relevant kinetic

equation in each order is found in terms of an expansion into

harmonic tensors (see appendix C):

f(c) = f(0)(c) +

+e2(f(2)(c) + f(2)(c).c + f(2)(c):<cc>) +

+ (1-2)

where c is the peculiar velocity, <cc> is the harmonic tensor

of second rank and f denotes an isotropic correction of

order n. Nonisotropic parts give rise to expressions for the

transport coefficients, isotropic parts appear in the contribu-

tions of the nonisotropic part.s in higher order. The expansion

generally used in the litterature is a two-term expansion of

the form:

f(c) = f(0)(c) + f(1)(c).c , (1-3)

which is sufficient for the calculation of transport coeffi-

cients in lowest order. The method applied in this thesis gives

results up to second order in e and describes both fast and

slow transport phenomena by means of the multiple time scales

formalism.

-5-

References

1. S .Chapman, Phil.Trans.R.Soc,216(1916)279,217 (1917) 118,

Proc.Roy.Soc.,A98(1916)1.

Q.Enskog,Inaugural dissertation,Uppsala 1917.

2. S.Chapman and T.G.Cowling:"The mathematical theory of non-

uniform gases",Camgridge University Press 1970.

3. J.O.Hirschfelder,C.F.Curtiss and R.B.Bird:"Molecular theory

of gases and liquids",.J.Wiley 1954.

L.Waldmann:"Transporterscheinungen in Casen von mittleren

Druck",in:iïandbuch der Physik, Springer 1958.

CCercignani:"Mathematical methods in kinetic theory",

Plenum press 1969.

J.H.Ferziger and H.G.Kaper:"Mathematical theory of trans-

port processes in gases", North Holland Publ. Comp. 1972.

4. G.Sandri,Ann.Physics,24(1963)332,380.

E.A.Frieman.J.Math.Phys. ,4^1963)410.

J.E.McCune,T.F.Morse and G.Sandri:Rar.Gas Dynam.J_(1963)115.

5. A.H.Nayfeh:"Perturbation methods", J.Wiley 1973.

6. P.P.J.M.Schram,:"Kinetic equations for plasmas",

Ph.D.thesis Utrecht 1964.

7. L.Spitzer and R.Harm, Phys.Rev. ,89_( 1953)977.

8. B.Davydov,Phys.Zeits.der Sowjetunion,8Q935)59 .

9. M.J.Druyvesteyn,Physica,1£( 1930)61, j_( 1934)1003.

10. W.van de Water,Physica,85C(1977)377.

11. M.Mitchner and C.H.Kruger:"Partially ionized gases",.J.Wiley

1973.

-6-

In order to describe a partially ionized gas one needs at

least three kinetic equations. Henceforth a plasma is

considered which consists of one-atomic neutral particles, ions

and of course electrons. Ionizing collisions assure the

presence of charged particles, but will just as the other

inelastic collisions be neglected when determining the distri-

bution functions for calculations of transport coefficients. If

the plasma is close to equilibrium one may use Sana's equation

to calculate the electron density from the electron

temperature. When the departure from equilibrium is larger, for

example because of radiation losses, it is assumed that the

electron density has been determined by other means. Thus the

collision terras in the Boltzraann equations of the three compo-

nents consist of a sum over all possible elastic collisions

that may.occur:

3f ,— s + vVf + — F «V f = ) J (f ,f ). (2-1)3t - s m -s v s L. st s' t x

s t=e,i,a

The left-hand side of this equation gives the total time

derivative of the distribution function of particles s under

the influence of a force F , for example external forces or as™ s

a result of a self-consistent electric field- The right-hand

side of equation (2-1) describes the variation of f caused bys

all possible elastic collisions.

Macroscopic quantities appear as so-called moments of the

distribution function f . Important quantities are:s

the density n , the hydrodynamic velocity in the laboratorys

frame w , the temperature T , the pressure P and the thermal-s s =s

heat flux <j . These are defined as follows:

n (r,t) = It (r,v,t)d3v, n w (r,t) = /vf (r,v,t)d3v,s - s - - s-s - - s - -

Pg(r,t) = /mgcscsfg(r,v,t)d3v, SB<r,t) = A n ^ c ^ f g(r,v, t)d

3v,

- 7 -

f n k T ( r , t ) = Am c^f ( r ,v , t )d J v , (2-2)i S S ™ S S S " ~

where the peculiar velocity c = v - w .—s — —s

If equation (2-1) is multiplied by appropriate functions of

velocity and integrations over the entire velocity space are

performed one obtains so-called moment equations. Choosing

these functions as: 1, m v, and ^m v2 the moment equations ares - s

the conservation equations for the particle number density,

momentum and energy respectively:3n

j^-+ V-(ngWg) = 0, (2-3a)

3w

s s 3t -s -s -s s-s s~ t*s S t S C (2-3b)

= Am v2{ [J (f f)^v,S t*s St S C (2-3c)

In the energy equation the following notation was introduced:

C = I kT + \m w2. (2-4)

The conservation equation for the particle number density is

called the equation of continuity. Equations (2-3b) and (2-3c)

are also frequently called equation of motion and of energy

respectively. In the right-hand side of these equations the

term corresponding to t=s disappears because it represents

collisions between identical particles for which the above

functions of velocity are collisional invariants1"2. Physically

this means that there is no net exchange of momentum and of

energy between like particles. One could have simplified

equation (2-3c) further with the aid of equation (2-3b) and

have arrived at the following form of the energy equation:

, 3T4 n k{-r-£ + w «7T } + 7«q + P : Vw » ftii c 2 I J d3v, (2-5)2 s l3t -s s ' a s - s - s J s s L st 't*s

- 8 -

a result that can also be obtained directly from equation (2-1)

with the velocity function \m c2. Another quantity of impor-s s

tance is the mass-velocity or plasma-velocity defined as:

£ m n w

s s s

It is possible to define diffusion velocities U with respect

to this plasma-velocity:

U := w - w . (2-7)-s -s -in

In a weakly ionized gas (WIG), however, the density numbers of

the charged particles are small. It follows that the mass

velocity almost equals the hydrodynamic velocity of the neutral

component. For later use diffusion velocities u are defined:

u := w - w . (2-8)-s -s -a

Now return to equation (2-1) and consider the right-hand side

of this equation. It consists of a sum of collision integrals

describing the variation in time of the distribution function

f due to elastic encounters only. One can distinguish two

different types of interaction: one based on a short-range

intermolecular potential and one of the Coulomb type, which

varies as l/r, r being the distance between two interacting

particles. The first of these applies to all collisions between

charged particles and neutral particles and between neutral

particles mutually, and will be described by the well known

Boltzraann collision integral:

J s t ( f s , f t ) = 2/d3M386<l2«1.*>{f8Cr ^ )ft(v+S+ ^ ) +t s t s

-fs(v)ft(v+g)}. (2-9)

Here g = v - v is equal to the relative velocity just before a

collision. The validity of the Boltzmann collision integral is

based on the stnallness of the number of particles in a sphere

-9-

with radius equal to the characteristic range of the potential,

i.e. the Boltzraann parameter. The notation in (2-9) is such

that it shows the integrations to be performed explicitly.

Indicating post-collision variables with a prime, the veloci-

ties just after a collision read:

ra Jl m S.vf = v — ~̂ ~"™" , v l = v + g + — — — , (2—10)— — m +tn —t — a m +nr

s t s t

where I = g1- g denotes the difference in relative velocities

just before and after a collision. The factor I(g,JL) is the

differential cross section and is defined as:

rl . (2-iD

where b is the impact parameter and x *s t n e scattering angle.

It contains the geometry of the collision. The 6-Dirac function

with argument A2-l-2g• Ji assures energy conservation.

Collisions between charged particles are more difficult to

treat because of the 1/r potential. The Landau collision

integral^ will be used, which can be obtained from the

3oltzraann collision integral in the impulse approximation,

based on the assumption that collisions change the velocity

only slightly. But one can also derive the Landau integral

directly from the well known BBGKY-hierarchy. The Landau

collision integral reads:

8!££ i iJ (f ,f ) = C V •ƒ[—- )•{- 7 - - V }f (v)f (v )d3v .st s' t st v n 3 ' lm v m v ' s - t -t t

Ê S C t

(2-12)

For reasons of simplicity only the velocity dependence of the

distribution functions in equations (2-9) and (2-12) has been

indicated. The constants C are given by:s c

q2q2lnA

To s

where q and q are the charges of the collision partners andS L

-10-

inA is the so-called Coulomb logarithm. Herein A is the inverse

of the plasmaparameter e , and is proportional to the number of

electrons in a sphere with radius equal to the Debye lenght r :

A = 7 - ner3. (2-14)

P

In a plasma one distinguishes three characteristic lenghts: the

Debye length A^, which is a measure of the distance over which

the potential of a charged particle is shielded by the surroun-

ding charged particles, the mean interpartlcle distance r and

the Landau lenght r , which is the distance of closest approachLi

between two like charged particle? with thermal velocities.

These lenghts are defined as:

rL = 7Tx> ro=*-1/3' ' D = £ T A C2~15)

One can verify that the plasma parameter is proportinal to the

ratio of the Landau- to Debye lenght, but also that the plasma

parameter connects all three characteristic lenghts in (2-15).

The condition for these lenghts to be well separated is that

the plasma parameter should be very small. The plasma is then

called ideal.

The Landau collision integral results after making two cutt-

off's: in the derivation of this expression there appears an

integral over the interaction distance diverging at zero and

infinity. The approximation made is that one introduces the

lenghts r and r as integration boundaries. This leads to the

factor lnA. This factor has to be much greater than unity.

Speaking in more physical terras one could say that the Landau

lenght is so small that there are relatively very few short

range collisions. Because of the effect of screening the upper

boundary can be replaced by the Debye lenght: collisions with

larger impact parameter contribute little to the collision

integral.

Next the electron Boltzraann equation will be considered in more

-11-

detail. To solve this complicated equation an expansion into a

small parameter e will be used, e being the square root of the

electron-atom mass ratio:

e = <me/maA (2-16)

This choice seems obvious and the next step is that all

dimensionless numbers, obtainable from the dimensionless

electron Boltzraann equation, are expressed as powers of e. The

equations will, however, not be made dimensionless. All terras

will be multiplied by the appropriate power of e. The distribu-

tion functions will be expanded into e and in the end e is put

equal to unity, so that e merely plays a bookkeeping role. For

a weakly ionized gas the electron Boltzraann equation reads as

follows:

3f eE-r-̂ - + evVf - e — «V f - u) (vxb) «V f = eJ + J + eJ .,9t - e m v e ce - - v e ee ea eie

(2-17)wherein b is a unit vector in the direction of a constant

external magnetic field B. The electron cyclotron frequency:

ui = — , has been taken of the order of the electron-atomce m

ecollision frequency:

w T =0(1). (2-18)ce ea

Here T is the mean collision time between two successiveea

collisions of an electron with a neutral atom:

v— = v = n v _ C r ' = r—£ . (2-19)T ea a Te^ea X v 'ea ea

Thermal velocities are defined as v =(kT /ra ) and Q is theAS S o S C

elastic collision cross section for momentum transfer of

particles s with particles t defined as follows:.. . IT

Q s t (g) = J2ita(g,x)(l-cosx)sinxdx, (2-20)o

where g is the relative speed of the colliding particles.

-12-

In expressions like (2-19) some characteristic value for g will

be substituted e.g. v . Furthermore the mean free path A hasle ea

been introduced.

The electric field has been scaled in such a way that the

energy gain of an electron in this field between two successive

collisions with neutral atoms will be compensated on the

average by the energy loss as a result of these collisions.

Then the following order relation holds:

e Te

Concerning the inhomogeneitits the Knudsen number ex defined as

the ratio of \ to some macroscopic length scale L reads:

xp0 (2-22)

where the ordering is in accordance with equation (2-17). The

order of magnitude estimation of the collision terms on the

right-hand side of equation (2-17) depends on the degree of

ionization and the kind of interaction. Because of the long

range of the Coulomb potential the Coulomb collision cross

section for momentum transfer is about 104 times larger than

the electron-neutral cross section. Coulomb collision cross

sections are defined on the basis of a 90° deflection. This is

necessary because of the weak interaction. Scattering is the

result of many grazing encounters.

A weakly ionized gas is defined such that the ratio of the

electron-electron to electron-atom collision frequencies equals

e:

J v n v_ Q ( 1 )

ee H ee e T ^_ H = _ ^ _

ea ea n v„ Qa Texea

The same holds for the electron-ion collision integral.

A strongly ionized gas will be defined as a plasma in which the

collision frequencies satisfy the conditions: v < v = v ..^ J ea ~ ee ei

-13-

Next the heavy particle Boltzraann equations have to be

considered. For a weakly ionized gas one obtains:

3f—«-.3t

- E2V • Vf =a ae

h Jaa

£2J , , (2-24)ai

3f eS-r-1 + E2v7f. + E2(— + ID ,VXb)'V f, = e \ j . + E J, + E2J . . .3t - i m c i - - v i ie ia ii

(2-25)

Some extra assumptions have been made in these equations. The

time variable has been scaled with T , so that in these equa-ea ^

tions the choice v = v = v /E has been made. The heavyea aa ia

particle electron collision terms receive an additional factor

e2 because of the fact that momentum transfer in these colli-

sions is rather inefficient. From these assumptions it followsthat Q *- EQ - Q. , which is reasonable provided that

xea xaa 4ia ' r

charge transfer is not taken into account.

At the same time it is assumed that the temperatures of the

different components are of the same order of magnitude, so

that v = v ~ EV . In the chapters to follow solutions ofTi 13 Te

kinetic equations will be found by means of a perturbation

expansion:f= (E.Y.O = fo

0)(E»Y,t) + ef ^ r . v . t ) +.... (2-26)s s s

It is known that such an expansion nay often lead to secular

behaviour, i.e. it contains terras f ,. and f such that thes,n+i s,n

ratio f ,/f goes to infinity with increasing time, sos,n+l s,n

that the expansion fails. One possibility to avoid these

secularities is to make use of the multiple time scale forma-

lism1*"7. For that purpose it is observed that there are

different time scales to be distinguished: tg is called the

fastest time scale which is connected with the mean free time

between two successive collisions of an electron with an atom;tn = T . Then successive time scales are defined in the0 ea

following manner: t1 = tQ/e, t2 = tQ/e2 etc. The t2 time scale

will appear to be the timescale on which energy relaxation

-14-

between electrons and atoms takes place. In the multiple time

scale formalism new time variables T are defined as follows:n

rn := ent, (2-27)

so that the time derivative transforms as:

IF * l?0+ 'h^^h^ <2~28>

Thus the formalism consists of a transformation from one time

variable to a certain number of time variables T which aren

treated as independent. In this way extra freedom is created,

that will be used to eliminate the secularities which may

occur. This is the essence of the multiple time scales forma-

lism. The expansion (2-26) then transforms as:f (r,v,t) + f (r,v,x ,T ,..) + f (r,v,x ,T ,..) +... (2-29)

The procedure is then as follows: the collision integrals are

also expanded in powers of £ and the expansion (2-29) is

substituted into the Boltzmann equation. Terms of equal power

of e are collected and equated to zero. The resulting equations

are then solved for the functions f . The conservation equa-

tions will be treated in a similar manner, and will serve to

find solutions to the kinetic equations. Substituting the

resulting solutions into the general expressions (2-2) trans-

port coefficients are obtained, mostly as integrals over the

electron-atom cross scetions. For realistic cross sections

numerical integration schemes have to be resorted to.

-15-

References

1. S.Chapman and T.G.Cowling:"The mathematical theory of nc

uniform gases", Cambridge University Press, 197

2. J.H.Ferziger and H.G.Kaper:"Mathematical theory of

transport processes in gases",North Holland

Publishing Company, 1972.

3. L.D.Landau,Phys.Zelts.der Sowjetunion,10(1936)154.

4. G.Sandri,Ann.Phys.24/1963)332,380.

5. E.A.Frieman:J.Math.Phys._4(1963)410.

6. J.E.McCune,G.Sandri and E.A.Frleman,

in Rar.Gas Dynara.j_ (1963)102.

7. G.Sandri,in:"Nonlinear partial differential equations",

ed.W.F.Ames, 1967.

-16-

III VERY WEAKLY IONIZED GASES

In the first chapter several categories of plasmas were

distinguished on the basis of the degree of ionization. In this

chapter the case of a very weakly ionized gas is considered.

Here the degree of ionization is so low that the effect of

Coulomb collisions is relatively small or even negligible. The

latter case has been considered by van de Water1. In the

following two sections a similar type of analysis is given for

a different ordering of some parameters. Inhomogeneities are

now assumed to be of the order e, whereas the influence of the

background neutrals is reduced as compared to his work.. The

ordering is then identical to the one used by Bernstein2.

Ir this chapter only the electron component is considered. The

distribution function of the neutral atoms is assumed to be a

local Maxwellian, of which the macroscopic quantities satisfy

the Euler equations.

In the third section the form-relaxation of the zeroth order

electron distribution function in a homogeneous plasma is

described for an arbitrary electron-atom cross section. This

differs from van de Water's work, in which also an inhomo-

geneous plasma is investigated but then restricted to a Maxwell

interaction between electrons and atoms.

In the last section collisions between charged particles are

included. The ratio of electron-electron to electron-atom

collision frequency is assumed to be of the order e2. The

influence of the electron-electron collisions on the electron

distribution function is nevertheless large. The form of the

zeroth order electron distribution function Is shown to be

governed by a non-linear integro-differential equation. The

asymptotic form of this equation describes the competition

between a Davydov and a Maxwell distribution function.

-17-

III-l The electron distribution function

In the Boltzmann equation for the electron distribution

function in a very weakly ionized gas only electron-atom

collisions are to be considered. Only the term J is thusea

retained in the right-hand side of equation (2-17). The heavy

atoms possess a local Maxwellian:

fa(r,Y,t)3/2

exp{-Y-wa(r,t)

3

[> (3-D

where the macroscopic quantities obey the Euler equations:

3n

3t+ V-(nawa) = O,

3w( —3 i /

f- (n T"3/2) - 0.dt ^ a a

The Mach number is assumed to be of the order unity:

M : =wI-a IV T a

= Oil).

(3-2)

(3-3)

(3-4)

(3-5)

From equation (3-3) the instationary inertial term is estimated

by means of the pressure terra:

V2

= 0{-T^)' (3-6)9t

VP,n na a

If the electron and atom temperatures are of the same order and

a velocity transformation is applied according to:

v -»• c = v - w (r, t),- - - -a -

the electron Boltzmann equation takes the following form:

(3-7)

3f«

3t

eE 3wec«Vf + e2w «Vf - { e - = - + e 3 f - r ^ - + ( w «7)w ) + e 2 ( c«7 )w— e —a e m dt —a —a — —;

+ eto w xb}«7 f - to c«(bxV f ) = J (f ),ce-a -' c e ce- - c e ea e

a

(3-S)

-•18-

where the ordering indicated earlier appears explicitly.

The solution of this equation is sought in the form of an

expansion of f in the small parameter e. At the same time the

multiple time scale formalism is applied; cf. chapter II.

The expansion of the electron-atom collision integral can be

found in appendix A. In zeroth order the following equation is

obtained from (3-8):

3f(0)

(3-9)ce- v-

It is possible to derive an H-theorem from this equation. In

velocity space a spherical co-ordinate

system with c directed along the unit

vector b is introduced. See fig. 3-1.

Equation (3-9) then reads:

3f(0) (0)

3fwce W (3-10)

fig. 3-1.

Multiplication of this equation by

(l+ln(f )) and an integration over

the entire velocity space results in:

3H(0)

W%f<°>ea

)d3c < 0

(3-11)where the inequality is proved in appendix B. Thus it is seen

that the zeroth order electron distribution function relaxes

towards an isotropic function when T Q + », since that is the

general solution of the equation J (f) = 0.

The first order part of equation (3-8) reads:

3TI " O

eE

J (3-12)

In a formal procedure one may separate the distribution

-19-

functions in an asymptotic part on the tQ time scale and a

remaining transient part:

f(O) = f(o) (0) (0) = llm f(o)#

e e,as e,t • e.as T + „ e

Then equation (3-12) is integrated with respect to TQ:

3f(0)

eE' T 3f(0)

-cVf +!i.Vf(°> +jC0)(f(D ,}+ /°{- fe.t _ c.Vf(0)e,as m c e,as ea e,as 3T 1 - e,t

c ^ < t ?Ve!t ea^t

where E' = E + w xB. (3-15)

If it is assumed that the integral in this equation remains

finite when Tg+ °», the first part in the right-hand side would

increase without bounds with T Q except if it is demanded that:

af(0) eE,

- e' a s+ c.7f

(0) - 5 . .v f<°> = j(°>(f(1> ) + u c-(bxV f(1) ).3T - e,as m c e,as ea v e,as' ce- y- c e.as'

This equation can be solved easily if f is expanded in

Insertion of this expansion in equation (3-16) then gives with

the aid of appendix A and definition (4-61) for M, .:=(n)

I {( l - na) bx) f ( 1> ( c ) } . ^ = J M . ,• f ( 1 ) ( c ) - ^ 1 ^ =, T^ v(c) ce~ n-e,asv Jn - L =(n) n-e,asv 'n -

n«l (n) n

g/' 3 8 » 0. (3-18b)

The latter equation is the isotropic part of equation (3-16).

-20-

Frora the right-hand side of (3-18a) it appears that n=l gives

the only contribution apart from an isotropic function f satis-

fying the homogeneous equation:

Jea ) ( f ) + uce£'(*xVcf) = °* ( 3" 1 9 )

Thus the solution for f reads:e,as

eE'9f(0)

fCD ( c ) = fd) ( c ) + T ( C ) C .M-1 . ( ^ _ e ' a s - Vf(0> ).e,as - e,as (1) - =(1) vm 3c e,asv

(3-20)

In second order equation (3-8) yields:

»f(2) 8 f

( 1 > 3f ( 0 ) ,,, . E. ,,,

For reasons of simplicity this equation will be dealt with in

the limit i •*• «>. The isotropic part can easily be separated

from the rest by means of the othogonality property of the

harmonic tensors (see appendix C) :

(0) ~(1)^e.as + £e,as + c^.^l) _ ±_ . a ( 3f(l) } + (0)« 2 ox1 3 -e,as „ 2 3c1- -ejas-* -a e,as

3f(0)

- | -.^V.w = J^)(f(0> ). (3-22)3 oc -a ea e,as

This equation may be integrated over T^, if w is assumed to be

stationary on this time scale. This is in accordance with the

Chapman-Enskog theory of the heavy particle gas. Then the

following equation results from elimination of the secular

terms:

3f(1>

= 0. (3-23)

With the results in appendix A and expression (3-20) for f-e,as

equation (3-22) is written as follows:

-21-

,f(0)

if;"

m ca

3

where <A' = V -*— . This equation has been derived earliere

by Bernstein2 and 0len3. An isotropic correction Is not

mentioned by these authors. The non-isotropic part of (3-21)

when TQ-»- °° reads:

, n ... eE'«c ... eE' 3f ( 1 )

c-Vf(1) +<cc>:7f(1) - ^ ^ f ^ ) - <cc>: - ^ ^ e ' a s

- e,as — -e,as ra c e,as — m c oc

3f(0)

-<cc>:Vw ~—-e'as = ai c«(bxV f ( 2 ) ) + J ( 0 ) (f ( 2 ) ). (3-25)-a c 3c ce- - c e,as e,as e,as v '

(2)Insertion of an expansion like (3-17) for fv ' leads to the

e,asfollowing solution of equation (3-25):

3f(0)

~(2)where the isotropic part f is as yet undetermined.

e j as

The third order part of the electron Boltzmann equation (3-8)

is:

„(0) „(1) „(2) „(3)! \ ^ ' < 2 ) < l ) <0>

dw- T^a«V f(0)- (c-Vw )-V f(1)- Ü) c(bxV f

dt c e - -a c e ce- - c e

where IF = IF + v 7 -The isotropic part of this equation can again be separated from

-22-

the rest. When Tg+ • this isotropic part yields the following

equation for the first order isotropic correction:

3f(0)e,as e.as

3~(2)e.as

eg1

3m c 2

e

3T,4

3» —3c

f

-e,as-a

?

e,as

c e3 3c

-e,asJea

fe.as

Insertion of expression (3-26a) for f

then gives:

(0) ~(1) ~(2)

(2)f clS

3f

TT,e,as e.as

3T1 -a e.as

(3-28)

and using appendix A

ofc e,as_

— — —— ' V*w —3 3c -a

^ f- e,as ^ 1)

kT

m cf }.e,asJ

(3-29)

Equation (3-29) may be Integrated over T^. Elimination of

secular behaviour then leads to the following equations:

3 f

3f

(2)e,as

1

(0)e, as

(3-30a)

3ff(1)

e,as

-a e,as

3c -xj1 .<Af

(1)=(1) - e,as

3fC_ ;

3 3c

m ca

( 1 )

—a

TÜ)

kT

m ce

fe,as

(3-30b)The latter equation for f is almost equal to equation

(0) e > a s

(3-24b) for f , which is homogeneous. Equation (3-3Ob) has a

source terra containing the zeroth order.distribution function.

These equations are different from the corresponding equations

of van de Water1, due to the different ordering.

The inhomogeneity of equation (3-3Ob) obstructs the absorbtion

of the first order isotropic correction into the zeroth order

distribution function, which was an assumption made by

Bernstein2. The equation for f is of second order in the

variable c. In the following section two conditions will be

given which determine the two constants of integration.

-23-

IIT.-2 The electron macroscopic equations

The macroscopic equations for the electrons can be

obtained from equation (3-8) through multiplication by the

appropriate functions of velocity and subsequent integration

over the entire velocity space. The following equations are

then obtained:

3n— e 0, (3-31)

3u dwm n f-r— + e(u 'V)u + e2(w «V)u } +e2m n (u »V)w + e3m n -r—e e'3t -e -e -a -eJ e e -e -a e edt

eV«P + een E + m n Ü) (u + EW )xb = Jm cJ (f ,f )d3c, (3-32)=e e- e e ce -e -a - e- ea e a

dln f-r-e + eu »V£ + e2w «Vê 1 + eV«(q + P «u ) + een u «E +p^-at- — o p - a e ' -»e = e - e P - P -

dw-a

+ era n u *(w xb) + e3m n u —r— + e2fp + m n u u ) : 7 w +e e-e

= A m c2J (f ,f )d3c.e ea e a

(3-33)

Note the transformation that has been made according to (3-7).

Therefore t is now defined slightly different from (2-4) as:

£ e = -| kTe + ^meu|. (3-34)

The macroscopic quantities are also expanded in powers of e and

the multiple time scale formalism (MTS) is applied. From the

above balance equations the following equations are obtained in

zeroth order of e:

(0) aï(0)3n

(0)3u

In first order one obtains:

3n on /rt\ /n\^ e f ><0>_u

m c

0,

(3-35)

(3-36)

(3-37)

-24-

„ ( 0 ) { ^ +!£e +( U<°).7)4O )

+ Io c e41>xb} + 7.P<

0)+ enf >E'

„ . 3u ( 0 )

'(1)

_ƒ_£!_ f(Dd3 (3_38)JT (c) e

+ en^0)E'«u^0) = 0. (3-39)

When T Q + «> the zeroth order electron distribution function

relaxes towards an isotropic function of velocity as was shown

in the previous section. This means that in this limit the

diffusion velocity u and the heat flux £ vanish.

Equations (3-37) and (3-39) take the following form when T.-»- =>:

9n(°> 3T<°>_e,as = _e,as = Q> (3_40)

en(0) (E'+ u(1> xB) + VP(0) + f^L-,f ( 1 ) d3c, (3-41)e,as - -e,as - r e a s 'iAc) e a s

where P <0 ) = n(°> kT

e,as e,as e,as

Thus it is seen that many terms in these equations vanish when

TQ+ a». The expression for f found in (3-20) may be

substituted into equation (3-41) which then yields an identity.

The second order equations are given in the limit T Q+ « in

order to reduce the complexity of the equations:3n 3ne a s _ e , a s +

a (1)

(0) (1)

-e,as e.as^1 vae,as 6e,as -e

d T ( ) ( )3 (0) (^e,as ^e.as (1) (0) j+ (1) (0) (1)z^eas kar 3T -eas eas^1 vaeas 6eas - e ^

-25-

+ en<°> E - ^ + P( 0 ) :Vw - - ̂ J l + ~*" f)f(0)e,as- -e,as =e,as -a m -'T, x(c) ̂ ra c 3c •* e,as

a (L) e

(3-44)

From equations (3-18a), (3-20) and (3-24a) it is inferred that:

3u ( 1 ) 3n(1> 3T ( 1 )

_^e,as = _e,as _ _e,as = Q< ( 3_ 4 5 )

It will be assumed now that the following first order quanti-

ties are zero:

n ( 1 ) = T ( 1 ) = 0 , (3-46)e,as e,as

which are the additional conditions needed for a unique solu-

tion of equation (3-29). Such conditions can in fact be chosen

without loss of generality on the basis of the arbitrariness of

the expansions of the initial conditions in powers of e. Since

moreover f is isotropic the second order equations nowe,as r n

°. (3-47)

= 0, (3-48)

reduce to:

8ne,as (0

(0) (2)en u xB +

e,as-e,as -

3 (0) rdTe,asIne,asKidT2

e,as-e,as -

) ( u ( i , +

m c

T ( 1 ) ( c ) f

• + P( o ) ve,as

w )

e,as

(0)e,as

- a

.( (1) + (0) (1)e,as-e,as'

(3-49)

As all quantities occurring here are functionals of f , seee g as

equation (3-20), (3-26) and (3-29), it appears that these equa-

tions do not contain any variations with T,, SO that the t.time scale has no physical meaning in this particular situa-

(2)tion. Insertion of expression (3-26) for fv ' into (3-48)

G y 3 S

leads to:

/c2<A'f(1') d3c = 0. (3-50)

This equation can be further evaluated to give:

-26-

Vp ( 1 ) + en ( 1 ) E' = O, (3-51)*e,as e,as- '

which ts satisfied through the requirements (3-46).

With the aid of equation (3-47) the energy equation can be

written in the following form:

- p(0) D ln{n(0) (0) 3 / 2 } + (1) (0) E, + (0) +

e,as D T 2 L e,as e,as ' -e,as e,as- re,as

k T . ,_.(1+ -Ji.|-)f<0> d 3 c . (3-52)v tn c 3c' e,as

where: £_ = f- + u(1) -V = f- + (w + u(1) ) -V.DT dT -e,as 9t2 -a -e,as

At this point it is suitable to introduce transport

coefficients. The first order electron diffusion velocity can

be calculated with the aid of expression (3-20):

n(0)u(D =-Iö(1>.E'+V.(n<°>D

(1)), (3-53)

e,as-e,as e = k e,as= ;'

3f(0)

where g ( 1 ) = - 3 ^ ; CT ( 1 )(c)^/ )-^e' a Sd3 c, (3-54)

S(1)-^öT-^2\i)fc>B?i)O3c- (3~55)

e,as

are the conductivity and diffusion tensors respectively.

If the solution of equation (3-24) for the zeroth order

electron distribution function is known, the transport

coefficients can be calculated. In a simple theory the

following approximation is often made:

f(0) = n(°> f (c), (3-56)e,as e,as 0

where f (c) depends on c only, so that the space and time

dependencies occur through n solely. With this assumption aG I 3S

diffusion equation may be obtained from equation (3-47):

^ 8 0 > s , . 0 i ( 3_ 5 7 )

where the neutral component has been assumed to be homogeneous

-27-

in space. The assumption in (3-56) also implies a uniform

electron temperature. Refinements can be obtained by making an

expansion of f in the spatial derivatives of n : see1 e,as e,as

e.g. reference 4.

These equations are used for the determination of electron-atom

cross sections from diffusion experiments5.

The thermal heat flux is also calculated with the aid of

expression (3-20):

3( 1> = - S

(1).E'- X(1).Vln(T(0) ) + V(n D ( 1 ) ) , (3-58)ae,as aq - = v

e,as' v e,as=q " v '

W i t h :

we,as e,as

5kT(0)

(1) = _ ^ £=

^ J T c V l f C ) d 3 c . (3-61)6 J (1) =(1) e,as

(2)It appears from expression (3-26a) for f that corrections

to the transport coefficients are given by the same expressions

if f<0) is replaced by f(1) .e,as e,as

From equation (3-50) one may infer then that in the special

case of Maxwell interaction between electrons and atoms the(2)

second order diffusion velocity u vanishes. The second™G j aS

order thermal heat flux reduces in this special case to:

3(2) = M-! 7J(2) j(2) . _ V 0 i ; 4~(l) d3c>ae,as =(1) q ' q 6 ' eas v '

The first order fluxes reduce to the following expressions in

case of Maxwell interaction between electrons and atoms:

kT C 0 )

-28-

showing that there are no cross effects In this case.

In third order of e the moment equations, when considered

asymptotically on the to-time scale, read:

3n(0) a (2)

e,as + _^,as + 7.(n<°>g_u<2>s) - 0, (3-64)

_

e

, (1) (2)(0) { 3 , a s + ^ e , a s ( 1 ) + (1) (1)e,as l3T2 3TJ v üe,as -aJ -e .as K-e,as }-a

+ V«P + en E + m n u (u + w )xb ==e,as e,as- e e,as ce -e,as -a -

m m c kT , , . . .= _£ /{.-ÊZ _ «CA ( _L_) _ k T 7 ( J L _ ) }f <D d 3 c ( 3 _ 6 5 )

ma T ( l ) 2 ~ C T ( D l c t ( l ) 6 l "

3£(0) 3Ï ( 2 )

(0) ,^e ,as + ffe.as (2) ^(0) } (2) (0) (2)ne,as^3T 3 ^ ye,as y°e,asJ ^3e,as Êe,as - e , a s ;

+ en ( 0 ) u ( 2 ) .K' + . n ( 0 ) u u ( 2 ) . ( . xb) =e,as-e,as - e e,as ce-e,as x-a -

m 3kT me 2

kT c i ^ -L- ) )^ 1 ) d3c. (3-66)a ^ T ( D

e'as

Again an Ansatz Is made, namely:

which can be justified In the same manner as in (3-46).

Equations (3-64) and (3-66) may then be written as follows:

3n(0)

eas -eas a e a s m 'x,.. m c

From (3-23) and (3-26a) It can be deduced that:

-29-

jf'as = 0. (3-70)

Equation (3-65) may therefore be written as follows:

,n. Du^ Dw ...(0) r -e,as + -a, (2) =

e e.as^Di^ Dt J =e,as

= j-!*! {1- flT(1) i-fc1* |-(J—))}f(1) d3c, (3-71)3 V ( 1 ) 2mec^

3c 9C T(l) "e'as

The survey of the moment equations has now been carried out up

to third order. The equations of this chapter are useful in the

process of solving the kinetic equations.

In the following section the equation for the zeroth order

electron distribution function will be solved in a special

case.

-30-

III-3 Form relaxation of the electron distribution function.

In this section the equation for the zeroth order

electron distribution function is examined for the case of a

homogeneous plasma without a magnetic field.

Equation (3-24) then takes the following form:

af(0) *

_ e> a s = A i_(-£Lfi + __s. i_)f(o) , (3_72)3T, jacW,, l^ucac^e.as'1 ( 3 7 2 )

2 m cz (1) e2* m

ale E T(l) ( c ))a ( l ) )

in which: Ta = Ta + < , (3-73)e

is a function of c. The relevant macroscopic aquations read:

"37e 'a S = 0, (3-74)

(0) p *3 (0) , 3 T e , a s rae,mec , _ k T a 8 , . ( 0 ) , 3-^e.as^, = -—/r~(1+i-c-^c-)fe,asd c- <

2 a (1) e

Equation (3-72) may be solved by means of the method of

separation of variables. Insertion of

f ( 0 ) = n ( 0 ) f(c)h(T2) <e,as e,as 2

into equation (3-72) results in the following eigenvalue

problem for the function f:

2 h 7 ^ ^ £ °' (3"77)m c' (1) eaand a simple equation for the function h:

~ + Xh = 0. (3-78)a T2

If X=0, equation (3-77) can be directly integrated. The

solution y0, the eigenfunction for \=0, then reads:

c m c 'dc 'y0 - A exp{- ƒ — ; }. (3-79)

o kTa(c ')

-31 -

This is the asymptotic solution of (3-72) when T 2+ °°, and is

known as the Davydov distribution function^. It will be demon-

strated now that all other eigenvalues are positive.

Define:

f(c) = yo(c)<j>(c). (3-80)

Substitution into (3-77) and subsequent multiplication by <j> and

integration then leads to:

(3-81)

ƒ yo(c)<)>2(c)c2dc

c2kTwhere yo(c) and p(c) = ^—r- (3-82)

aT(ir

are positive functions, so that all eigenvalues except X=0 are

positive indeed. Expression (3-81) also gives a device for the

calculation of the eigenvalues and eigenfunctions by means of a

variational principle. From equation (3-77) one can deduce that

all eigenvalues are orthogonal with weighting function c2yQ:

Jyoc2(j) (j> dc = 0, n+ra. (3-83)

o

The variational principle then reads as follows:

= min R(<f>) = R(<t>n); |yo(c)<|>n(c)<j>m(c)c2dc = 0, m=0,1,... ,n-l.

(3-84)

where: R(<|)) = — . (3-84a)

Jyo(c)<t)2(c)c2dc

o

A Rayleigh-Ritz method may be used to approximate the first N

eigenvalues and eigenfunctions. In the special case of Maxwell

interaction between electrons and atoms the eigenvalue equation

can be solved directly. Then the collision time T. . is a* (l)

constant, so T does not depend on c either. The eigenvalueSi

equation after a transformation of variables reads:

-32-

m w2

where w = j- . Equation (3-85) is the differential equation2kT

a

of Laguerre. The eigenvalues and eigenfunctions are thus equal

to:

y = 4 n )(w), A = | 2 - , n=0,l,2 (3-86)n * n T(l)

The Davydov distribution function is now a Maxwellian with*

temperature equal to T .3

In the case of a hard spheres interaction model one has:

T ( 1 ) ( C ) - | , (3-87)

where I is a constant mean free path. A straightforward calcu-

lation shows that the Davydov distribution is now equal to:

2 . m m (AeE)2

y0 = C exp(-ac2)(l + j~) , a = 5--=- , A = -2 (3-88)

a 3m2kTe a

where the constant C is fixed by:

00

/yo(c)c2dc = 1. (3-89)

0

In the cold gas limit T •»• 0, the Druyvesteyn distribution7 is

recovered:3m3

y0 = C exp(-Yc't), y = . (3-90)

4m (AeE)2

a

If the eigenvalues and eigenfunctions are known, the initial

value problem may be solved, i.e. equation (3-72) supplemented

by the condition:

The formal solution reads:

-33-

with: a => Jfo(c)<J> (c)c2dc, (3-93)

o

if the eigenfunctions are orthonormal:

J*2(c)yo(c)c2dc = 1, n=0,l,2,... (3-94)

o

and form a complete set.

The same problem has been investigated by Braglia et al8, who

calculated the temporal behaviour of the distribution function

for various cross sections.

-34-

III-4 The inclusion of Coulomb collisions.

In the foregoing sections the Coulomb collisions have

been neglected entirely. If, however, the electron density is

such that the ratio of the electron-electron to electron-atom

collision frequency is of the order ra /m , i.e.:e a

n 0 'a ea

the e-e and e-i collision terras appear in the second order

equation of section 1. When T Q + <=°, only the isotropic part

changes, and the equation for the zeroth order electron distri-

bution function now becomes a nonlinear integro-differential

equation:

»jlO,l+ (0) c*jlO,ls 1_ , 3 . 1 , (0) ,3T -a e,as 3 3c -a 3c - (1)=(1) - e.as"1

2

, me 3 r c3 ., a 3 .-(0) •> ,r(0) ,(0)

m c 3c lx n. m c 3c e,as ee e,as e,asa (l) e

In third order of e the results of section 1 change as follows.

To the nonisotropic part of the electron distribution function

terms proportional to c are added coming from the Coulomb

collisions and the equation for the isotropic correction in

first order becomes of the same type as equation (3-96). In

order to study the nature of equation (3-96) this equation will

be considered in the special case of a homogeneous plasma

without a magnetic field. With appendix D-l one obtains:

or £.\J « /rw / ̂ \ "̂-*- oi.ni r J

e,as ee 3 t A U ) (c)\n ' fii e,as e,as •>3TT ~ -> 3c^ e,as^ ' L e,as*- m c ~ 5c '

2 m c z e

J ( v c ) v f = (v) (

m ca

-35-

The asymptotic solution of this equation may be considered as

the result of the competition between a Maxwell and a Davydov

distribution function. Omitting the time derivative and

integrating once one obtains the following equation for the

asymptotic solution f :A

2C « 31n(f (v)) 31n(f (c))\-s—j(v3~có)vf. (v) T r [dv +

m l3 J A lv 3v c 8c 'e c

kT 31n(f ) m c3 kT 31n(f )n . f l + — - — r — — ) ] + — fl+ — T — — ) = 0 , (3 -39 )

A*- m e 9c Ji m T , , . 1 1 m c 3c ' ' v '

e a (1 ) ewhere the constant of integration has vanished by consideration

of the limit c

following form:

of the limit c + °°. The equation for f can be written in theA

IT W

*

T.JL-ËI) + v (i+ ÈL) = o, (3-99)T. dw ee dw

Am w 3 / 2 m e 2 2C n

where B(w) = 2/2 — - r-rr , w = s p _ v = e e A

m aT ( 1 )(w) 2kTA ee ^kT m 3/2

V T = > fA = nA ( y T 7 ) exp(y(w)). (3-99a)

i e e A A ZTTKIA

The following normalizations should then be imposed on a

solution of equation (3-99):

/exp(y)w%dw = — , /exp(y)w3/2dw = ̂ ~- , (3-100)o o

in order to determine the integration constant and the

temperature T . If w » l , then the solution of (3-99) may beA

approximated by the solution of the following equation:

A

The solution of this first order differential equation is:

-36-

w B(w') + vy(w) = - ƒ{ 5E ££ }dw' + C, (3-102)

o B(w')T /T. + va A ee

where the Integration constant C and the temperature T^ areA

fixed by conditions (3-100).

The problem has been Investigated earlier by Lo Surdo9, who

obtained solutions for simple electron-atom cross sections by

means of an iterative numerical procedure. It seems that,

because equation (3-99) is of a simpler form than his equation,

the results of this section might lead to simpler numerical

techniques to obtain a solution.

-37-

References

1. W. van de Water, Physlca 850(1977)377.

2. I.B.Bernstein, in: Advances in plasma physics vol.3 (1969)

3. A.0ien, J.Plasma physics, 26^(1981)517.

4. L.G.H.Huxley and R.W.Crompton, "The diffusion and drift of

electrons in gases", J.Wiley (1974).

5. H.B.Milloy et al, Austr.J.Phys. 30(1977)61.

6. B.Davydov, Phys.Zeits.der Sowjetunion £(1935)59.

7. M.J.Druyvesteyn, Physica 10(1930)61,1(1934)1003.

8. Braglia et al, II nuovo cimento 62B(1981)139.

9. C.Lo Surdo, II nuovo cimento 52B(1967)429.

-38-

IV WEAKLY IONIZED GASES

In chapter II a weakly ionized gas (WIG) was defined as a

plasma in which the ratio of electron-electron to electron-atom

collision frequencies is of the order e (cf. equation (2-23)).

This means that the degree of ionizatton is very low. Since the

Coulomb collisions become more important at lower temperatures

the degree of ionization should be assumed to decrease with

temperature in order to satisfy the ordering mentioned above.

In this chapter the procedure is as follows. Firstly the heavy

particles are considered, because they can be treated as almost

independent from the electrons, i.e. as a binary mixture.

Because the degree of ionization is low the usual Chapman-

Enskog equations are only slightly modified. Then the electron

Boltzmann equation which gives more interesting results will be

dealt with. The isotropic correction to the zeroth order

Maxwellian electron distribution function is not adequately

dealt with in other theories, with the exception of van de

Water's paper^. It also appears in references 3 and 4, but does

not receive the attention it deserves. The expansion of the

electron distribution function in powers of e leads to some

results which are not found with the usual harmonic tensor

expansion-*. .

The isotropic correction results from the competition between

the mutual electron collisions which try to establish a

Maxwellian and the disturbing effect of electric fields,

temperature differences between electrons and heavy particles

and temperature- and pressure gradients.

The domain of the degree of ionization in a WIG can be roughly

devided into two regions. At lower degrees of ionization the

isotropic correction is important whereas the corrections due

to multiple collisions dominate at higher degree of

ionization. Expressions for the electron transport coefficients

will be derived and finally the modifications in case of a

seeded plasma are given.

-39-

IV-1 Heavy particle results

The heavy particle Boltzniann equations valid in a WIG

were already given in chapter II, equations (2-24) and (2-25).

The distribution functions are expanded in powers of e and the

multiple time scales formalism (MTS) is applied. Up to second

order the results are:

( 0 ) 3f ( 1 )

l- + ! l a =J ( f V l + J (f (1 ) ,f (0 )), (4-2)3TX 3 T 0 aa a ' a aa a a '

f(0) 3 f(D af(2)

^ = 0, (4-4)

af(0) 3 f(D

< <l) «i 2 ) (0) «5 (0)I72 +Tt\ + T F ' +Y.Vf[O) + ( +̂<,clvxb).7vf5°> =

(0) (1) (1) (0) (0) (0)Jia(fi 'fa } + Jia(fi 'fa ) + Jii(fi 'fi )# < 4" 6 )

By means of an H-theorem obtainable from equation (4-1) it

follows that f ' relaxes to a Maxwellian when t n+ "•• This

a u

limit will be indicated by a subscript "as" so that:

(0) . n(0). n((0)

_

a,as a,as

In order to proceed the moment equations are needed. The

balance equations for the neutral particles read:

-40-

3n-^ + e2V-(nawa) = 0, (4-8)

3wm n (^a + e2(w -v)wJ + e2V-P = e2jm vJ (f f )d3v, (4-9)3 3 OC ~3 3 ~3 3 3x 3 1

(4-10)

in which the interaction terms between the heavy particles and

the electrons are omitted because these are of the order e1*.

The macroscopic variables are also expanded in powers of e and

the MTS formalism is exploited. Up to second order the results

from these equations are:

3n(°> 3g(0> 3w (°>

< < <

3n ( 1 ) 3n ( 2 )

< TF; i%i°> 0. (4-13)

3w(0) 3w<^ 3w<2> „.

; a ; 4 0 ) ^

From equations (4-7) and (4-12) and the definition (2-4) of

chapter II it is concluded that -r^ f£ - 0. Then equation (4-2)

becomes in the limit T.+ °°, indicated by a subscript "as":

J (f<0) ,f<1> ) + J (f ( 1 ) ,f(°> ) - 0. (4-16)aa a,as a,as aa a,as a,as

This equation possesses the following general solution2:

-41-

a.as " vul T *2mï T U 3 V ^ a , a s ' V H i / J

where o.(r,TltT2,...) are at this point arbitrary functions of

position and time. The Chapman-Enskog choice:

n(D = W ( D = T ( D = Qa,as -a,as a,as

makes these functions zero, so that the first order correction

to f vanishes:a,as

f ( 1 ) = 0. (4-19)a,as '

Next equation (4-5) will be considered in the limit T Q+ °°:

T F = Jia(fi as'a as'1 ( 4" 2 0 )

This equation also possesses an Il-theorem implying that £.

relaxes to a Maxwell distribution function, when T Q+ ", with a

temperature and a hydrodynamic velocity equal to the neutral

ones:

- „(0) "

T ^

aA

A subscript "A" denotes the limit Tj+ «. The ion balance

equations read:3nj^ + E2V.(n1?i) = 0, (4-22)

3w.

e/m±yJia(f1,fa)d3v, (4-23)

(4-24)

After expansion in powers of e and using the MTS formalism the

results up to second order of e are:

-42-

3n<°>

a (0) (1)

(4-27)

(0) (1) (2)3n. 3n 3n+ ' + J

i l ci-i

(4-30)

When T.* 00 f^0) is a Maxwellian with w[°^= w( ^ and T ^ = T^K1 1 -IA -aA iA aA

Furthermore the first order corrections n.. and T., areiA iA

assumed to have vanished. Equations (4-29) - (4-31) then read:JO)^ * ,.c.2>.S') • 0. (

-43-

where p£^= n^'^aA^ and -'= - + -aA>x-* (*"35)

if n /m = 0(1).With the definition of the total derivative:

fc/fc/sS»-. '«')equations (4-32) and (4-34) can be written as follows:

< + n(0)^(0) = 0>

dw(0) '(O)-aA (0)_ (0) _ " v '"̂

miniA "dT2 piA eniA -

(4-39)

And for the neutrals the Euler equations are obtained:

.(0)

. (C)

C°>

When equations (4-39) and (4-42) are compared with each other

it appears that there is no net energy exchange on the t2-time

scale between ions and neutrals in first order:

which is compatible with the choice T^, = 0.1A

-44-

Now equations (4-3) and (4-6) can be treated. When TQ* "° these

equations read as follows:

3£(0)

aA

which are the Chapman-Enskog equations for f . and f.. . The

left-hand sides of these equations can be brought into a more

familiar form through a transformation in velocity space:(0)

v •>• v - w . crom the laboratory frame to a frame moving with— — aA

the zeroth order hydrodynamic velocity of the neutrals.

With the aid of the macroscopic equations (4-37) - (4-42) the

source terras of equations (4-44) and (4-45) become:

(4-46)

aA aA

"i -nu ,rvv eE'

aA aA aA

(4-47)

where c = v - w . , which is the peculiar velocity defined in

chapter III. The equations (4-44) and (4-45) are consistent

with the traditional Chapman-Enskog procedure, see e.g.

Chmieleski and Ferziger3.

If one considers the heavy particle results of reference 3 in

the case n « n the equations (4-44) and (4-45) are recovered

with source terras (4-46) and (4-47) respectively. The solution

of these equations is standard2. If resonant charge transfer

instead of elastic scattering is the main mechanism for the

-45-

ion-neutral interaction the zeroth order ion distribution

function will in general not be a Maxwellian. When a constant

cross section for the charge exchange process is assumed the

deviations from a Maxwellian are not very large, even in the

absence of ion-ion collisions5.

-46-

IV-2 The electron Boltzmann equation

The electron Boltzmann equation for a WIG has already

been given in chapter II: equation (2-17). Contrary to the

heavy particle equations a transformation in velocity space

from the variable v to the variable c = v - w will be started- - - -a

with. Equation (2-17) then reads:

3f-^e + e(c + ewa)-7fe- oi^c-CbxV^) +

eE 3w- { e — + eu w xb + er^a + e2f(c + ew )«v)w 1-V f =1 m ce-a - 3t kV- -a J~a' c e

e

where b is a unit vector in the direction of B and w = — is

ce m

the electron cyclotron frequency. The hydrodynaraic velocity of

the neutrals has been taken of the order of the thermal veloci-

ty, i.e. the Mach number is of the order unity. The electron-

heavy particle collision integrals are expanded in powers of e,

the velocity variables are assumed of thermal order. The

results are presented in appendix A. In the expansion of J

the first order term vanishes because of the transformation in

velocity space mentioned above. After substitution of the

expansion for e and exploiting the MTS formalism the results

from equation (4-48) up to second order are:

e , e . _ „,(0) r -

D c«(bx7 f̂ ;̂ce- - c e

-47-

(0) (1) (2)

"• + ii- +ii- +c.w<i>+»«

o>.vf<o>-. (w(i)xb).vf(°

3 T 2 3TJ dTQ - e -a e cev-a - c e

a (0) . (1)_ ( f ! + u w(0)xb).? f(D_ & +!£a +c.Vw(0)}.vf(0)+

''ia c e - a -' c e <*Ti ^ T o - - a - ' c e

3 ( 0 )

1

(4-51)

From equation (4-49) an H-theoretn can be derived, see also

chapter I I I , :

,„«»e < 0, (4-52)

° H< 0 ) = ft ̂0)ln(f ̂0))d3c, (4-53)

so that again the zeroth order distribution function relaxes to

an isotropic function when T--> °°. In that limit one obtains

from (4.50):

3f(0> , „ eE' ( 0 )

e a s ( 0 ) ^- e» a s + c.(7f(0) - - ^ T -e ' a s l - u (bx7T, - ̂ e,as m c 3c ; ce^- fc

J<°>(f(1) ) + J (f<0) ,f ( 0 ) ), (4-54)ea e,as ee e,as e,as ' v '

where J . (f ) vanished because of the isotropy of f .ex e,as e,as

Isotropic and nonisotropic parts of equation (4-54) can be

readily separated, see appendix C, so that the following

equations are obtained:

3f(0)

ILe.as = (0) (0)3TJ Jeeue,as'£e,as;'

U c ( b x V f ( 1 ) ) + J ( 0 ) ( f ( 1 ) ) - c ^ f ( 0 ) (4-56)ce- - c e,as ea e,as e,as v '

eE' 3where: cA' = V -r- , as in chapter III. Equation (4-55)

-48-

also permits an H-theorem which states that f relaxes to ae,as

local Maxwell distribution function when T,+ °°:

(0), me'eA (

A

m c'e

2kT(0)•}•

eA eAA solution of equation (4-56) can easily be obtained if fis developed into harmonic tensors (see appendix C):

(4-57)

(0)e.as

f (c) f(1) (c)«c + f(1) (c):<cc>e,as -e,as - =e,as —

f(1) (C)KCS ,n~e'aS n "

(4-58)

where f is a tensor of rank n and • denotes an n-fold dotn-e,as nproduct.Insertion of this expansion into equation (4-56) gives:

n=l l(l)'- no> bxl f(1) (c)^.<c^ = lM

ce- ;n-e,as Jn - => b fce- ;n-e,as

n=i/1N. f

(1) (c)-<cn>(1) n-e,as n -

where : 2irn c/o(c,x) (l-P (cosx) )sinxdx,a Q n

(4-59)

(4-60)

(see appendix A).

If b is directed along the z-axis M, . in index notation reads:

M — o — no) T (c) £ b » (4—61a}(n)ij ij ce (n)v ikj k'

M,nü)ceT(n)(c)eikjbk

(4-61b)

Only the first two terms in the expansion of f are non-zero

so that the solution of (4-59) is:

e.as

where f

i 44(1) 2? (4-62)

6j asis a yet undetermined isotropic contribution. It is

in fact the homogeneous part of equation (4-56). In much the

same way equation (4-51) will be treated. When TQ+ °° the

isotropic part of this equation reads:

-49-

df<°>df 3f 2 , „ 3f ^ e E a , „e,as e,as c^ (1) _ c e.as (0)_ - 3 ( 3 (1)

di 2 9Tj 3 -e,as 3 3c -a 3ra c^ 3c*- -e,as-

= -!S- [JEL ( l +fa_ 3 }f(0) + (0) ?(D +o LT,1X

V- m c 9c ' e , a s J ee e ,as e ,asm c^ (1) e ' ' 'a

+ J ( f ( 1 ) ,f ( 0 ) ) . (4-63)ee e ,as e ,as

This is the Chapman-Enskog-like equation determining the first

order isotropic correction. The non-isotropic parts give the(2)

following solution for f in the same way as in the case of6 y 3S

the first order part:

, _ ( 0 ) (0 ) ( 0 )/ T \ i / n \ i C . n . u , of

f(2) - V l ,M: ; , . { -^ f ( 1 ) - e l 1>a s- i>a s —e>a

-e.as (1)=(1) l - e,as ^ 3c

2C ^ Q )

- e i i'aB f ( 1 ) + .J (f ( 1 ) )}, (4-64a)^ -e.as 1 ee -e.as >'

m e 3 '

3f ( 0 )

f<2> - x„,M:ï,.l-^f(1) -lJ-e'aS7w<°> }, (4-64b)

=e,as (2)=(2) l - -e,as c 3c -a,asJ

where the constant C is defined in appendix A.ei

Again there appears a yet undetermined isotropic contribution.

In equation (4-64a) the following linearized electron-electron

collision term was introduced:

J ( f ( 1 ) ) : = J ( f ( 0 ) , f ( 1 ) ) + J ( f ( 1 ) , f ( 0 ) ) =ee v e , a s ee e , a s e , a s ee e , a s e , a s

This expansion is justified because the collision operator is

rotationally invariant in velocity space.

In expression (4-64a) for the correction f the contribution-e,as

of the first order isotropic correction appears. The last two

terms between braces express the influence of the Coulomb

-50-

collisions on the electron-atom interaction and are referred to

as the effect of multiple collisions. The first term of these

may lead to divergent expressions because of the factor c~3. It

becomes even worse in higher order terms. In appendix E It is

shown that one can replace m c3 by [m c3 + 2C .n^ T/i\(c)] *n

6 6 cl 1)3S ^ i j

the denomerator of that specific terra. This is actually an

improvement because it results from renormalization of that

term.

The foregoing procedure can be continued up to arbitrary order,

but it will not be done here. The higher order equations can in

principle be solved, but the increasing complexity impedes the

actual calculations to be done. When T.+ °°, it has been deraon-(0)

strated that f . is a Maxwelllan and a solution of (4-63) can

be constructed. Before doing so the electron balance equations

will be dealt with first.

- 5 1 -

IV—3 The macroscopic electron equations

The moment equations for the electron component of a WIG

can be obtained from equation (4-48) by the normal procedures.

With the definitions of the diffusion velocities u (see-s

chapter II) and the definition c = v - w one can see that:— —a

— Set (r,v,t)d3c = w - w = u . (4-66)

e

The electron balance equations for a WIG provided with the

appropriate powers of e then read as follows:3n-^e + eV-(neue) + e

27-(ngwa) = 0, (4-67)

3um n f-̂ -e+ e(u «V)u + e2(w «7)u } + eV«P + een E +e el3t -e -e -a -e' =e e-

3w+ emenehiïa + e ( - e ' V ) - a + e 2 ^ a " V ) -a^ + meneuce (-e+ e - a ) x -

= /n?e^^Jea(fe'fa) + e J ei ( f e ' f i} ^ 3 c ' ( 4 " 6 8 )

3 5 en (-T- + eu •?£ + e2w «V£ ) + eV«(g + P «u ) + em n u *(w xb)

ev.gt -e e -a e' -*e =e -e e e-e - a -

3w+ een u »E + em n u •f-rr- + e2w «Vw 1 + e2fp + m n u u 1: Vw =

e-e - e e-e '•St - a -a.' l=e e e-e~eJ -a

= Amec2(jea(£e,fa) + e^^f^fpjdSc. (4-68)

Where now, slightly different from equation (2-4):

le = ̂ e + ̂ meUl ' (4'7O)

All macroscopic quantities are expanded in powers of e and

again the MTS formalism is used. In zeroth order the results

from equations (4-67)-(4-69) are:

3n(°> 8*<°>

(4-72)

-52-

And in first order of e:

_ « + ._« + ,.<„<%<<»> . 0, (4-73,

. (0) , (1) , < = )

& 1 ^E * . „ » (u „ > b • . „< l )

U u(

e - e e ce -e -a - e e ce-e

s L f(D d3 c _ 2C .„(O)jJ.el 3c3

e„(0)E.u(0) + u . „(0)»<0).(»(0),b) - 0. (4-75)e - -e ce e e -e -a -

Here and in the sequel the results are used that were obtained

in preceding sections, e.g. —- w' '=0. When T + <*>, equations

(A-73)-(4-75) can be further simplified, because then f ^ isCO) (0) e'aS

isotropic, which implies: u = q = 0 etc. In this limit-e,as -*e,as

the first order equations become:

8n<°> 3T(0)

TF^ S = < * a S " 0, (W6).(1)

7p̂ 0^ + en(0) E + « n

W M (u(1> + w

(0) ) xb + h^H^c =0.re,as e,as- e e,as cev-e,as -a,as - JT n.(c)

(1) ( W 7 )

Substituting the expression for f as given in equation6|aS

(4-62) obviously renders equation (4-77) into an identity.

Further observation shows that equation (4-77) 'closes' when

T. .(c) does not depend on c, i.e. the case of Maxwell Inter-

action. The electron-atom interaction potential Is then assumed

to vary as r"1*.

Equation (4-77) in case of Maxwell interaction reads:

s^l)~e,as *

(4-78)

o> ( ) b

e e,as ce-e,as -e,as -

- 5 3 -

which is the generalized law of Ohm in first order.

In second order the balance equations will be considered

asymptotically when TQ* ", then they read as follows:

8n(0) 3 n(De.as e.as { (0) ( (1) + (0) ̂

3 T 2 3T 1 *• e,asv-e,as -a,as ' '

^ T ( 0 ) 3T3 , r e , a s L e,as ̂ (1) ̂ ,(0) , ̂ (0)3 , r e , a s L e,T ne,ask{HT2

+ TT,

+ V . ( 3( 1 )

+ P ( 0 ) . u ( 1 ) ) + P ( 0 > :Vw ( 0 ) =-*e,as =e,as -e,as =e,as -a,as

m tn c2 kT^0) . . . .= _ _£ƒ_«_ (i+ _ £ i S £ 3 ) f ( ° ) d 3 c . (4-81)

m JTfl.v m e 8c' e ,as

The derivative T — was defined in equation (4-36) of section 1.

Now the following Ansaz is made:

(1) = T ( D = 0 (4-82)

e,as e,as ' v '

which will be verified in the next section.

The equations (4-79)-(4-81) then assume the following form when

Tj+ » (subscript A):

(0)

lï3 C ^ u ' " . <4-84>

.w

-aA

-54 -

Next the local entropy density in zeroth order is introduced:

Then it is possible to rewrite equation (4-85) as an entropy

balance equation:

3s , q . /, x ,„N ,-... i . *X + g „ »XeA (jeA (1) (0) (0) } = -eA -m ^eA -q

3T, „(0) -eA -aA ' eA ' m(0)TeA xeA

- ̂ V 1 m c 2^1 f e f(0) ,3 .,

Ö i TT feA d C' (4"

ÏTÖJ OÖ) TTeA eA

where the thermodynamic forces:

kT (0 )

K » - - | - t 5 f +-T i Vln(p2 ) )} (*-88«)e

X := -Vln(T(^) (4-88b)

have been introduced. The first terra in the right-hand side of

equation (4-87) gives the entropy production which is positive

definite. This may be proved by means of Schwartz' inequality

with the aid of expression (4-62) for f . . It also gives the

relations between the fluxes i . = m n . u . , q . and the-eA e eA -eA -*eA

forces as defined in (4-88a,b). These relations, which also

obey the Onsager reciprocity relations, read as follows:i(l) = . n(0)u(l) . m n(0) ( 2 0(l). x. + D ( D . X j-eA e eA -eA e eA *• Te = -m =T -q;

s ü ) - ' r l ^ 1 ) ^ + i(1)-V (4"89b)

in which the transport coefficients are tensors because of the

magnetic field. The subscripts "T" and "D" stand for thermal

diffusion and Dufour effect respectively. The expressions for

these coefficients are:

D ( 1 ) = -ifljy /T(1)(e)c2f2)H"1

1 jd'c. (4-90.)

- -f-JfeA =CO d ' (4~99b)"eA """eA

- 5 5 -

/1 \ kT * t n c c 2 / rw i

*( 1} • - r* ^(D^>c2(r!(ó)" 4) f e ^ a )ttcT ,

eA

The divergence term In equation (4-87) contains the entropy

flux, consisting of a thermal and a convective part. The last

term in this equation represents the entropy exchange between

the electrons and the neutrals.

Finally the third order equations in the limit T,* °° will be

given. Again an Ansatz is made:

which will be verified later on. In third order of c there

results from equations (4-67)-(4-69) when Tj+ °°:

3 (0)

A («»>+ ̂ ?>)xb = - A i f(3)d3C - 2Ce eA ce -eA -aA - rd> e^ e

+ —— ra tS°,K .u^ + —(vl 2 A'X1 + B«X ), (4-93)3 v ^ j - e iA ei-lA m^ Te = -m = -q •"

eA TeA

e/ aA .

^ 2 ?8 v f pi e i r

(?>e r e__r., aA a \v(l).3 , i n i \

The tensors A and 3 in equation (4-93) are defined as follows:

V 2 5 (0) -1 3 (A"95)

(1) 2fcT(0) 2 eA-(l)

-56-

kT ( 0 )

, , . 1 aA 9 r u 3 1 ïwhere: a(c) = - — ?rlc 'aT ~Z— J-

T(l) 2c1* 3 c 3c T(l)

In equation (4-94) the entropy exchange with the ions appears.

Because of the conditions (4-82) the first order part of the

entropy vanishes:

0. (4-96)

Then one may add equations (4-87) and (4-94) to obtain the

total entropy balance equation up to third order.

The entropy production term in equation (4-94) can be evaluated

using the expressions (4-62) and (4-64a) in the formulas for

the fluxes. It appears that those parts corresponding to the

multiple collision terms in (4-64a) give positive definite

contributions to the entropy production. This could have been

anticipated because these contributions depend linearly on the

forces defined in (4-88).

Another important conclusion that can be drawn is the follow-

ing: if T--. is independent of c great simplifications occur in

the momentum and energy equations, see e.g. equation (4-78).

In equation (4-94) the second term on the right-hand side which

contains the isotropic correction, vanishes because of the

conditions (4-82). Further it is observed also that in the case

of a constant Tf-i\ t n e cross effects are absent in first order.

-57-

IV-4 The first order isotropic contribution

In section IV-2 the equation for the first order

correction, equation (4-63) has been derived. When Tg+ «• f

is a Maxwell distribution function- The equation for the

isotropic correction f . then reads:(0) e A

2

£ 2 ? VeA

+ -S_J_f_£l f a -l If (0) 1 (4-97)a ( U eA

where:

e eA

The left-hand side of equation (4-97) contains the linearized

collision integral defined in (4-65) which is from now on to be

understood as follows:

Jee<f> mJM<''f™> + Jee«™-V- (4~99)

i.e. asymptotically on the t. time scale. The moment equations

(4-83) and (4-85) will be used to eliminate the time derivative

in the first terra in the right-hand side of equation (4-97).

The Coulomb collision integral can be written as a divergence,

see appendix D. When the following integral operators are

defined:

c °°

I (f) = *jj Jvp+2f(v)dv, J (f) = *£ /vp1"2f(v)dv, (4-100)

P CP 0 C C

it is possible to integrate equation (4-97) once. The

integration constant vanishes, as can readily be verified. The

result is then:

m lL o eA ' ,, (0)*- *K eA ' lv eA '}i eA

m e

-58-

e eA eA e eA

p(0) m <0)lT(0) LJix feA Ci « T l T(0) ^ e A

eA a eA eA eA

V.lJf2T(1)(c)M-1

1).{x;-^) + X q(-^L - f)f<°>}j. (4-101)

eA eA

It is then advantageous to make a change of variables from

(c,r,x2) to (w,r,T2) where:

m e 2 kT^?* , .w := -^TTT, , — ^ | - - f- , V > V + wX | - . (4-102)

2 k T ( 0 ) ' mec 3c 3w ' -q 3w

eA

The functional notations are not altered after this

transformation. Further the function g is introduced which is

related to £ , according to:

g(w) := (1+ — ) f ^ ' . (4-103)

Finally an integral equation for the function g is obtained:

iw 3/2°°

F(w)g(w) - j / x 3 / 2 g(x)dx - -*~- /g(x>d x = Kw>> (4-104)0 W

in which the source terra is defined by:

VTe

3m2 . T ^ !

J$) (0)eA neA

)ee eA

? ,(0) " e 7 ' ^ , Ame7ieA (0) +7^=(0) 7ieA (0) + 7 ^ = J0 T (x)

eA K1e m"7F ° U ;eA ma"7F ° U ; (4-105)

-59-

where: T (w) = —- , T = ( n ^ v Jl Q ) - 1 , (4-106)Q(w)/w a

w , w2 w 2 ( w - j )Gi<w> = TTT̂ T ' G 2 ( w ^ = w Q ( w ) . G3(w> = 7T7TX • Gu(w> = — F 7 ^ — »

G5(w) = eW J { 2 - ^ So - - ^ M-l }e-Xdx,

w 7,2v^ x ( x - f ) , , _xe J {—— Rn ~/_\ ïT7iTv l e <*x»

5. - x

The function F(w) is defined as follows:

F(w) := irV*erf(wS/4 - w^/2 , (4-108)

of which some properties are:

F(w) =Ö"(w3/2), w*0 ; |^(F(w)e"W) = ^ w V W . (4-109)

It can be verified that exp(-w) is a solution of the homo-

geneous part of the integral equation (4-104). The integral

operator is symmetric and real, thus exp(-w) is also a solution

of the homogeneous adjoint equation. Then it is required that:

GO

/e~Wb(w)dw = 0. (4-110)

o

This equation turns out to be the energy equation of the

electrongas. By means of a special operation on equation

(4-104) it is possible to obtain the following simple ordinary

differential equation for the isotroplc correction:

where the new source term J(w) is connected with b(w) in

equation (4-104) via the relation:

(4-112)

-60-

When J is put equal to zero in (4-112) a second order homo-

geneous differential equation is obtained for b which has two

solutions: b=constant and b=w^/2. This means that the second

part of expression (4-105) does not contribute to the final

solution. From equation (4-111) it follows that the general

solution for f . reads:eA

00 Ot>

f ^ = e~W ƒ ƒ eXJ(x)dxdw' + Cie~W + C2we~

W. (4-113)w w'

The constants C, and C2 are fixed through the requirements in

(4-82) leading to:

ƒ rt V'2<lw = ƒ fUAV

/2dw = 0. (4-114)J eA ' eAo o

Thus it was legitimate to make the Ansatz (4-82).

Again in the special case of Maxwell interaction between

electrons and atoms (T,,.= constant, i.e. Q(w) proportional to

w ) the source term b reduces to:

60 e ^ 0> [ {^ -« eA ))SÜJ11 eepeA

Only those parts relevant for the solution are given here. This

expression vanishes for a homogeneous electron temperature.

-61-

IV-5 Electron transport coefficients

With the results of the foregoing sections it is now a

matter of straightforward substitution to obtain the electron

fluxes. This section will be restricted to the case without a

magnetic field. This means that the tensor MT.1. becomes equal

to the unit tensor. As a consequence the tensors G5 and G_6, SQ,

RQ become also proportional to the unit tensor so that e.g.

G5 = G5Ï. The electron fluxes in first order then read:

a<l> := Am c2cf (1>d3c - &°y\} = Xn{v-2 RQX + L0X },aeA e - eA 2 eA -eA 0 Te 0-m 0-q'

(4-116b)

^T^ 4Tn(?)(kT(J))2 » w ( w 5 VeA , eA v eA ̂ T f v 2 .

D 0 " ~ ' X0 ' L0 = J Q 7 ^ — dw'in 3/7 m 3/7 o Wl ; ., ,,_.

e e (4-117)

Observe that 3 . is the thermal heat flux, defined in terms of

the peculiar velocity of the electrons, see also expressions

(4-89) and (4-90). In second order the electron fluxes become:i(P - /m cf(Pd3c = m n^Djgv-^-S , + — S )x +-eA e- eA e eA "L Tev ei «- ee;-m

/IT

' ^R" R ^ + ^ °l " ^ f ö )eA

eE 5k 3k T

where: B~ = (Ö")'{knvTe -ra + ( k 3 i " ~~F^~ 1 ± — ("7?) ~1 ^kT . m SL2 T .

eA a eA

5i eA -m , 6 i reA -q , 3i-m -q , , „o . . , „ „ .+ TTTV + (ft) ^ + ; — * + k

4 lx q ' (4-120)

eA Te peA Te

-62-

In the expressions (4-118),(4-119) the following coefficients

were introduced:

Sei = Iei(1)' Rei = Iei(w-!>' Lei = ^ « " " f ^ -

Ree = W1'*"!*' Lee

:= ƒ f<w>™fdw i (f ,g) = /f(„)/{g(w)}dw. (4-121)o Q(w){w2Q(w)+e} e e o

where £ is a linear integro-differential operator, see appendix

D. This operator also plays a role in the so-called Spitzer

problem, see chapter V. Also appearing in (4-118),(4-119) are:

3m T ( 0 )

£i = ( V i-off»»!+ *icf*ffi+ {k"^]" k"x<a eA eA

+ (k, - k,,)X «X - kc.(V«X - X2) - k,.V»X }x +6i 3i -q -p 5i -p -p 6x -qJ-q

eV(V«E)

4 i q 51l Si—W- + k517 ( VV +

eA

e(k - k ) V(E»X )

CO)eA

(4-122)

where: X := -V l n ( p ^ ) . (4-l22a)

A mean free pathlenght I has been defined by:

I '.= v TJ , v|e = kT^)/me. (4-123)

The parameter fj is of the order e and is proportional to the

e-e to e-a collision frequency ratio:

n(0)c

sLsse e e e 2m v3 e e Snem

e Te o e

-63-

The coefficients k. . appearing in the expressions above are

defined as follows:

00 0O CO

- w r r r x ,k := /e"W{ ƒ ƒ eXJi(x)dxdw' + C + C21wJH.(w)dw (4-125)J o w w'

w w _ dGwith: J.(w) = 4-[-=r-r f e x -j- dxl.

ï dwLF(w) J dx J

o(4-126)

The functions H. are defined by:

H 1 ( W ) = Q T W V H2<W> = f X ' H3(w> = ^

H^(W) = — K — 'ÜÜM , Ht(w) =ü_(wH, ). (4-127)dw b dw ^

Note that in general the coefficients k . depend on the cross

section and on the temperature as well; this as a result of the

definition of the variable w.

The complexity of the second order results makes It desirable

to restrict the calculations to a number of special cases. In

table (4-1) five different situations are specified.

1

2

3

4

5

E = 0

X = --P

X = --q

i "

a eA

Vlnl

Vln(

0

0

T eA ) }

* eS

+ eE

= 0

= 0

- k T ( 0 ) ( k X + XK eA U T - q -

- k T ( 0 ) r k x + xeA A-q

\ v — p / e

table

(4-1)

In each of these situations the expressions for the electron

transport coefficients are much simplified. In general these

are defined as follows:

-64-

,(2) _ % (2) (0)r (2)„ .n(2) •> (2)-eA e - e eA l -p T -qJ -ex '

Hl (2} (2\ (2\ (2\q , — — O h. + A_. X + A X + q ('t—J./OD)aeA q - T) -p -q aex

(2) (2)In these equations the second order fluxes I and q are

^ -ex ^ex

different from the other terms because they are not

proportional to E, X or X . They give no corrections to the

first order fluxes, but are new effects. Their general form is:

in which the vectors Y are defined as follows:

i kT(0) -^ kT(0) - -J kT(0) - -peA eA eA

eA

Y 7 = VXZ, Ya = V(V«X ) , YQ = 7(X «X ) . ( 4 - 1 3 0 )— / p — o —q * — y —p — q '

The ccifficlents a. and b can be expressed in terms of k :

a l ' k l i ' b l " km«

2 "" if i» 2 ~ 4 4 '

a 3 = " a 4 = a 5 = k 5 1 ' b 3 = " b 4 " b 5 = k 5 4 '

a 6 " k 6 1 ' b 6 = k 6 4 '

a 7 "~ 11 mm 51 ' 7 14 ~ 54'5 5

8 ~ 2 .11 3 1 51 ' 8 ^ 1 4 34 54»

a9 = k31 k61' b9 = k34 ~ k64" (4-131)

This section is concluded with some expressions for the

transport coefficients in some special cases mentioned in table

(4-1). The electrical conductivity in case 3, i.e. when no

-65-

teraperature gradient is present, reads:

eA eA

. /? T<°>(ö)7-I * k52<% ~ *?l + k22 4^(^ÖT "OeA eA

JUwhere:

aee p°ei"(4-132)

0 ^ • (4

3/7r e

The thermal heat conductivity in case 4, where there is no

first order electrical current, is:

k5H7 '5p + ^ 5 4 + k 1 4 ) x ] + - £ _ - K j - g - - i )

Ei L .

eA

7(Lee" Vee> " «L.i" V.i> ̂ (4"133)where:

K2 Y26 K T T 2 5 + 5Y24 K34

K D = Y, c - k Y-, c + ^Y^, ~ k,, ,3 36 1*35 3k 6 4 '

K4 = k26 " kTk25 ~ ^k24»

] f 4i T 1̂ 2 T l ~i T Li'

Y2i = 6i ~ r 5i 3i ~ i^li ~ T 11'

Y3i = k6i " kTk5i* (4-133a)

The thermal diffusion coefficient for the electrons up to

second order in case 4 reads:

-66-

m fü T ( 0 )

00 21 + k 2 3 ) ^ 4 ë ( - ^ "O - » a l + f Ree+

3 eA

+ • r T ^ l l c 5 X *X + K R V ' X + K 7 x 2 " k s i V # X + ( k m + k i i > x 2 l 1»120 ->-q - p 6 -q ~ q 51 - p 51 11 p ' '

(4 -134)

where:KS = Y23 " ^Y21 + k 3 1 " k S l 2 k

Tk l l »

K6 = Y33 " ^Y31 ~ k61»K? = Y13 " Kil ~ K l + k T k U ' (4 -134a)

Finally, again In case 4, the coefficient for the Dufour effect

for the electrons reads into second order reads:

A^>+ X̂ 2> = An{R0 + k25 ̂ 4 f e -1) - BR , + ̂ R +D D 0' ° 25 m 4B

V (0) J ei /- eeeA

All the foregoing results for case 4 can be transformed to

those of case 5 by simply replacing k_, by k •

Further It is observed that only those parts in expression

(4-134) and (4-135) that originate from the isotroplc

correction do not obey the Onsager symmetry relation.

Numerical examples are worked out in chapter VI.

-67-

IV-6 Modifications for a seeded plasma

Alkali seeded noble gases are of practical Importance for

MHD-generators, which operate at low temperatures. It is then

nevertheless possible to obtain a sufficient degree of ioniza-

tlon because the seed is easily Ionized. The partial seeding

pressure will be rather low, but the elastic cross section for

momentum exchange is rather high as compared with noble gas

atoms. Therefore the case where the electron-atom collision

frequencies of the noble gas and of the alkali seed atoms are

of the same order of magnitude will be considered. In the

electron Boltzmann equation a term J , (f ,f, ) is added where b

eb e b

denotes the seed. In the expressions for f one has to

replace x. . by T. . defined as:

a M bT + TT(D CD

( 4" 1 3 6 ;

Inserting the expressions for T.'. one obtains (see (4-10Ó)):

-aT

Where an electron-atom cross section for the seeded plasma has

been introduced. When the seed has the same temperature as the

neutral gas it follows that T /T = n./n i.e. proportional toD 3.

the relative seed concentration.

The collision terms in second order of c are also influenced by

the mass difference between the gas and the seed atoms. It is

not difficult to see that for an isotropic function f:

a s,ra e s,T

where:

-68-

Td) b T d ) Td) b a T d )

If the heavy particles are in thermal equilibrium these

collision times are equal. The first terra in the right-hand

side of equation (4—63) should be replaced by expression

(4-138) with f = f . In short the modifications to be made6 y ELS

for a seeded plasma are: replace evrywhere T.. . by expression

(4-136) except in the energy equation: (4-31),(4-85),(4-87) and

(4-94) where (4-139) is to be used. This is also necessary in

expression (4-97) where the energy equation has been used. The

last term in equation (4-97) should then read:

( 4 " U 0 )

m e a

where

*T = Ta

c 3

now:

(0) _aA

[ a 1TeA°

se maT(l)

)f ( 0 )] .

T T ( 0 )

s,m eA „,? - =(D Kl

1 Te

X'-m

(4-141)

The general formulas which are obtained up to now will be used

in chapter VI to calculate the transport coefficients for

actual situations with realistic cross sections. These calcula-

tions will be compared with the results of mixture rules and

with experimental results.

-69-

References

1. W. van de Water, Physica 85C(1977)377.

2. S.Chapman and T.G.Cowling, "The mathematical theory of

non-uniform gases", Cambridge University Press, 1970.

3. R.M.Chmieleski and J.H.Ferziger, Phys.Fluids 10(1967)364.

4. V.G.Molinari.F.Pizzio and G.Spiga, II nuovo cirnento

53^(1979)95.

5. I.P.Shkarofsky,T.W.Johnston and M.P.Bachynski, "The

particle kinetics of plasmas", Addison Wesley, 1966.

6. P.M.Banks and G.J.Lewak, Phys.fluids 11(1968)804.

-70-

V STRONGLY IONIZED GASES

A strongly ionized gas was defined in chapter IT as a plasma

in which all elastic collision frequencies of the electrons are

of the same order of magnitude. Then the parameter e is absent

in the right-hand side of the electron kinetic equation and the

solution of this equation should be valid for arbitrary degree

of ionization. Unfortunately this cannot be fully exploited in

practice for the following reasons.

Firstly the isotropic part of the electron distribution

function shows strong deviations from a Maxwellian as demon-

strated in the preceding chapters, whereas In the present

chapter it shows up in second order. In the second place the

polynomial expansion mostly used to approximate the solution

for the non-isotropic part converges very badly for low degrees

of ionization, especially in the case of argon because of the

Rarasauer minimum*. That is why the restriction has been made

that all collision frequencies of the electrons shall be of the

same order of magnitude, except for the fully ionized limit,

which can be taken without any severe problems. The fully

ionized case Is thus a special case of the results of this

chapter, as far as the electrons are concerned.

The equation for the non-isotropic part of the electron distri-

bution function for a fully Ionized plasma has been solved

numerically by Spitzer and Harm2. Sonine polynomial approxima-

tions were used by Landshof3 and Kaneko4 among others. With the

inclusion of a neutral species the problem has been attacked by

many authors5"9. In this chapter this problem is reconsidered

and it is shown that the equations can be written in the form

of a self-adjoint differential equation, which permits easier

calculations. The connection with the weakly ionized case as

treated in chapter IV is also demonstrated.

When the electron collision frequencies in the Boltzmann equa-

tion are of the same order of magnitude the following order of

magnitude relation for the densities holds:

-71-

7 (5-1)n Ql ;

The electron-electron collision cross section Q is much

m e

larger than Q : the electron-atom cross section. Therefore

the assumption will be made that the degree of ionization is of

the order e:•jp =<?(£), (5-2)

a

which, of course, implies a limitation for the validity of the

heavy particle equations. With the assumption (5-2) the heavy

particle Boltzraann equations read:

3f— a + e 2v7f = e3J + J + eJ . , (5-3)3t - a ae aa ai3f-r- + E2vVf. + e2^ (E + vxB)»Vf. = e3J, + eJ. , + eJ. . (5-4)dt — l m , — — i ie ii i.a

The right-hand side of equation (5-4) contains extra factors e

because the fastest time scale corresponds to the e-a collision

time. It has been assumed that the electron-atom and atom-atom

collision frequencies are of the same order of magnitude. Heavy

particle-electron collision integrals receive an extra factor

e 2 because of the inefficient momentum transfer process (cf,

equations (2-24) and (2-25) of chapter II).

The electron kinetic equation now reads:

3f eE 3w— e + ec«Vf + e2w «Vf - (e— + eu w xb + z-f? +e2(c«V)w +3t e -a e *• ra ce-a - 3t - - a

e

+ e 3 ( w »V)w ) . V £ - u> c ( b x V f ) =• J + J + J , , (5-5)-a -aJ c e ce- - c e ea ee ei*

where the transformation to the hydrodynaraical velocity of the

neutral gas has been made as in chapter IV:

c := v - w . (5-7)_ _ ~3

In the following sections a similar procedure as in chapter IV

will be followed. Firstly the heavy particles are dealt with;

after that the electrons.

-72-

V-l Heavy particle results

The heavy particle equations are only slightly altered

when compared with the weakly ionized gas in chapter IV sect.l.

Equations (4-1) and (4-4) remain unchanged so that f is a

Maxwellian as in (4-7). The equation of continuity for the

atoms is also identical to (4-8). The factor e2 in the right-

hand sides of (4-9) and (4-10) is now replaced by e. The

results in zeroth order from the balance equations are again:

3n(°> 3w<°> 3£<°>3. a a -. . — _ ,.

•57 = -rr = r- = 0 . (5-7)3 T 0 3T Q 9 T 0

The results from the continuity equations are the same as in

the case of a WIG. (Jp to second order they read:

3n<°> an*" 3n(°> 3n(1> 3n<2>

< - A ~~ °' < + < + < + V'(ns as0>> = 0, (5-8)where s=a or s=i. The macroscopic equations for the ions are

all the same as in the WIG, see equations (4-25)-(4-39)•

The momentum- and energy equations for the atoms now yield up

to second order the following results:

au(0) (1)

3»<0>

) ,(1) ,(2) ,(0) ,<l)

(5-12)

-73-

In order to proceed the kinetic equations have to be considered

simultaneously. From the equations (5-3) and (5-4) the follow-

ing equations are obtained in first order of e:

3f(0) a,(U£ • + " « .j ( f V ' j t J (f(1)f(0)) + J f(f<

0\f<0)>,3T. 3xn aa a a aa a a ai a ' 1 '

(5-13)

It is shown in appendix B that from these equations an(0)

H-theorera can be derived Implying that f. relaxes to a

Maxwellian, with a hydrodynamical velocity and a temperature

equal to those of the atoms, when T,+ °°:

f(0) <0>, "I , , il a l ,

Contrary to the case of a WIG the conclusion that f doesa,as

not depend on Tj cannot be drawn. From equation (5-13) it

appears that only when T,* °° the same equatTon for the firstorder contribution f . as in the case of the WIG is obtained:aA

J C f ^ f ^ + J (f^f^) = 0. (5-16)aa aA aA aa aA aA '

With the Chapman-Enskog choice:

it can be concluded that the first order correction is absent

if T,+ »:

^ = 0. (5-18)

The second order equations derived from (5-3) and (5-4) read:

3f(0) af(l) af(2)

f'. 3f(2)i _j

O

-74-

eE

mi

p(0)

_ (0) (1) (i) (0) (0)" Jia(fi 'fa )+Jia(fi 'fa )+Jii(fl

When Tj+ o» these equations reduce to the Chapraan-Enskog

equations for the corrections f.. and f . :

f{0)>.(5-20)

^ a A v.vf

3(0)aA

ff(0) f ( 2 ) ) + J

aa UaA • aA ; J(2) f(0)aA ' aA ; J

3f(0)

r(0) f(l)j

(5-21)

(0) (1)Jii(fiA 'fiA

(1) (0)Jii(fiA >fiA

(5-22)

In order to evaluate the right-hand sides of these equations

the balance equations have to be considered in the limit T,*

As far as the ions are concerned they are given by equations

(4-37),(4-38) and (4-39),which will be repeated here:

(5-23)

dw

The macroscopic atom equations read:

(5-25)

(5-26)

,(0)d2aA (0) (0)iA (5-27)

(0) 2

a I - -aA(5-28)

-75-

Addition of (5-25) and (5-28) gives with the aid of (5-23) and

(5-26):

From (5-25) and (5-28) it can then be concluded that there is

no energy exchange between Ions and neutrals on the t

scale in this order:

where c = v - w , a slightly different definition as used in

(5-6). This result is the same as obtained in case of a WIG(see

equation (4-43)).

The energy equations thus reduce to the Euler adiabatic

equations of state. Momentum transfer, however, does take place

on the t2-timescale. Addition of the equations (5-24) and

(5-27) gives:

5' = 0, (5-31)

where ph = m/»+ .f™. ph - p ^ + pg', Th = T^\ (5-32)

The left-hand sides of the equations (5-21) and (5-22) can now

be evaluated in terms of the macroscopic quantities. After a

transformation from the variable v to the new velocity variable

c = v - w one finally obtains:

aa aA ' aA ai aA ' iA

£ . { ( ^ - |)Vln(Th) + ^ l ] ^ . (5-33)naA

- 76 -

( f f ) + J Cf ( 1 ) f ( 0 ) )i i l t l A * iA ; J i a U t A ' aA ;

+ ^ ) 5 l a } ] f <J\ (5-34)h h n,

iA

where n, = nV. + n*. and the diffusion driving forces are

defined by:

n.. (ID — in )n.a n m n , n ,

-ia - a i »• r^ J n p ^ph

(5-35)

These equations can be seen as a special case of the ones

obtained by Chmieleski and Ferziger8. This is due to the

restriction made in relation (5-2). Their equations for the

heavy particles are coupled, whereas here equation (5-34) can

be solved independently for f.A . Substitution of the solution(2)

into (5-33) then gives an equation for f . The solutions canaA

be obtained by means of a traditional Sónine polynomial

expansion10.

The ion- and atom Chapman-Enskog equations (5-33) and (5-34)

are thus seen to be only weakly coupled due to the choice of

the specific domain of degree of ionization. The coupling

becomes stronger when the ion density increases. See also the

corresponding equations (4-46), (4-47) for the case of a WIG.

-77-

V-2 The electron kinetic equation

The kinetic equation for the electron distribution

function, equation (5-5) will now be treated along the familiar

lines. The zeroth order equation reads:

(5-36)

It is easily shown that from this equation an H-theorera can be

derived implying that the zeroth order electron distribution

function relaxes to a local Maxwellian when T_+ °°:

e,asexp{-

T

e,as2kT(0)

"I- (5-37)

e.as

The left-hand side of the first order equation is the same as

in equation (4-50), whereas the right-hand side now becomes:

( 0 ) ( f a )J ( 0 ) ( f a ) ) + J (f(e a e e e e

J (fe e e

+ Jei { e ; + Jei U e >£i >m

When T Q+ °° the first order equation reads:

(5-38)

eE.

<f ) + J ( f ) + J ! ( f ,fi ) ,ee e,as ei e.as' ei v e,as' i,as"

(5-39)

where J (f) is the linearized collision operator (see (4-65)).

In the next section it is shown that n and T do note,as e,as

depend on tp hence from (5-37):

TF,"e,as _

E 0. (5-40)

Then equation (5-39) becomes:

ce - c e,as - ea e,as e,as ei

-78-

p >e,as

„(0) C3e,as

where appeadix A and expression (5-37) have been used to

evaluate the right-hand side; u. Is a diffusion velocity,

see (2-8). The isotropic part of this equation simply reads:

J (f ( 1 ? ) = 0. (5-42)ee e,as

The function f is assumed to be expanded as in equatione f as(4-58). The general solution of equation (5-42) is:

HI TOT)e,as

see chapter IV section 4. In (5-43) A and B are as yet

arbitrary functions of space and time. The choice:

n ( 1 ) - T ( 1 ) = 0 , (5-44)e,as e,as '

makes them zero. Then the conclusion is that there is no first

order isotropic correction in a strongly ionized gas:

From equation (5-41) it appears that f is proportional to c

only. The magnetic field makes it necessary to separate the

components of f in the following way:

n(0)

-fe!L=-7̂ f Hf^ + f^+f^},, (5-46)

where: A = (b'A)b, A. = A - A.., A = b*A. (5-47)

The vector A stands for one of the vectors between braces in

eE'(5-41): Vln(p(0) ), Vln(T(0) ) , —^rrr- . »i°^ . The summationV ie,as" e.as'' ,_(0) -i.as

-79-

in (5-46) is over these different possibilities. The general

form of the equation determining f then reads:1 61 as

2C ,ii(0) B(0)

ai 6. f, +ce it i

ra c"e

1_ ie e .cis , , vJfj = ^QJ—*— b(c),ne,as (5-48)

where i = ll,l,t; the subscript k has been omitted and 5 Is

the Kronecker delta. If A = 71n(p ) then b(c) » 1, and so

on, see equation (5-41). Equation (5-48) will be dealt with

further in section 4. The operator J was defined in (4-65).1 ee

Without the term (T (c))-1f the equation is Identical to the

equation that has been solved numerically by Spitzer and Harm1.

In second order the electron Boltzmann equation has the same

left-hand side as equation (4-51) of the WIG. The right-hand

side now reads:

eaJ(2V°\f(0)) + J (ea e a ee

ee

( 0 )) (0))j ( fJei ( e j ( fJei U e ' (5-49)

In the limit T Q * <*>, the isotroplc part of the second order

equation reads:

df

f"*... 3kTvf ywe.as -a,as -e,as

e.as

(ea e,as e,as

f(0) p (1) . (1) + j ?(2)eas u ee -eas eas - ee easee -e,as e,as - ee e,as

\\fW -c.f5°> )ei -e,as - i,as (5-50)fi^hf ,fS ),° ei e,as i,as

where Pfl Is the operator which when operating on some function

gives the isotropic part of that function. In general (see also

appendix C):

1c

- 8 0 -

Equation (5-50) is an equation for the isotropic correction~f 2 }

f and is of the same type as equation (4—97) for the first

order isotropic correction in the case of a WIG.

When Tg* » the nonisotropic part of the second order equation

reads:3f( 1 ) eE' m e 3w( 0 )

e,as

+ J!f_ f(°> <cc>:Vw<0) - u, c (bx 7 f(2) ) =kT(0) e,as — -a,as ce- - c e,as

e,as

- J(0)(f(2) ) + J (f(2) ) + J«\fW ) + J™<fW .ff(1) )ea e,as ee e,as ei e,as ei e,as i,as

(2 )Next f is expanded into irreducible harmonic tensors. Then

e,as v

equation (5-52) can be separated into two equations: one for

-e^as atld one fGr =e2=the aid of appendix A.

The equatit

and reads:

f and one for f • The collision terms are evaluated with-e.as =e.a«=

The equation for f is of the same type as the one for f^ -e.as 1V -e.as

<2>(2)

o bxfce- -e,as

e,as

•s»< 0 )

e,as

(2)(2)The equation for f takes the following form:

=e,as

2u,c, bfce- =e,as

' ee,as

-81-

-e,as m c -e,as ._(0) e,as -a,ase,as

C n(0) ( 0 )

ei i,as-i,as{6 4 8 )f(l) _f (1) (1) _5

m lc° cH3cJ-e,as * ee -e,as - -e,as -

When T,+ <» these equations simplify further since then uj =0.~1 ) 3.S

Equations (5-50),(5-53) and (5-54) can in principle be solvedIf the first order contributions f\ and f are known. The

-i,as -e,as

equation for the latter will be discussed In section 4.

Contrary to the case of the WIG it was demonstrated in this

section that the equations for the isotropic and rconisotropic

parts of the electron distribution function are found in the

same order. It also appeared that there is no need for a first

order isotropic correction as In the case of a WIG.

-82-

V-3 The electron macroscopic equations

The moment equations in a strongly ionized gas will be

treated with the aid of the corresponding equations (4-67) to

(4-69) of the WIG. The only alteration to be made is to drop

the factor e in front of the e-i collision term in equations

(4-68) and (4-69). The zeroth order equations now read:

e e 1 9 T 0 ce-e -> ' l i 3 • ' e - e

In f i r s t order of e:

^ • w; + v.<.f'4»', - o,

(0) (1) (0)/ _ » Oil OU /r\\ / / - \ \ / i \ oU

n . (u w ) b . n ( 1 > . u ( 0 ) xb =e e c e - e -a - e e ce-e -

( 0 >

(0) £(1) m c

c f d c _ 2C ( 0 ) ; £e - e e i i J c J e

(5-58)

(5-59)

ral

terms in these equations to vanish. The equations then read as

(0)When T.+ °°, f has become isotrcplc, which causes several

6 « clS

-83-

follows:

a (0) .„(0)on oT

IT6'38 = IT6'38 = °' (5-60)

„ (0) . (0) (0) (1)Vp + en E + m n io u *b =e,as e,as- e e,as ce-e,as -

?r n ( 0 )

_ƒ(_!_ + ei i,as)m (l) d 3 c + _ 8 _ m n(0) y (0)T(l) m e 3 e~ e' a S 3/27 e e' a S ei-l'as

where: vfil = " [ ^ 0 ^ ^In contrast to the case of a WIG this equation does not 'close'

when T,.. Is Independent of c, I.e. the case of Maxwell inter-

action.

In order to reduce the size of the formulas the second order

equations are only given in the limit TQ->- <»:

9n ( 0 )

if'** + 7-(^1S4:L) = 0, (5-62)

xb +rf-2— + ei i.aSN- e.as r. . ei-e,as_

(5-63)

3 (0) {^e,as (1) (0) , + 7 . ( (1) + (0) (1) } +Te.as Idi, ae,as e,asJ ^e.as Fe,as-e,as'1

e,as-e,as - e,as -a,as ei i,as-i,as •* 'c3 e,as

or ™ n ( 0 ) t ^ 0 5 a_ /Leimenl,asr fl +

KIi,as _ ) f ( 0 ) d 3 c +

m1 J l c m c2 3c e.as

e

(0) _ 2

a ' a S . . e —+ k-T ; el-f-1—) If4 ; d3c. (5-64)

See appendix A for the evaluation of the moments of the

collision integrals. When use is made of the fact that f

a Maxwellian further simplifications can be obtained.

-84-

If the local entropy density is defined as:

it is possible to write the energy equation (5-64) in the

following way:

9 Q ( 0 ) .(I) , (1)

2 T ' ' ' Te,as e,as

(0) o (0)in T . ra c ... 8m v . ._. T.er a,as n -il f e „(O) 3 e e l (0)

7 ? ^ 5 r ?e,as e,as i e,as

- 2C ̂ i0) OJ°> •ƒ V0 d3c, (5-66)

ei i,as-i,as ' cJ e,as

where i , X' and X are defined, just as in chapter of the~6) 3S *"* ni "~T

WIG, as:

1^> :=mn<°> u^> , X' = - *• (E' + ^ L , P ^ ),-e,as e e,as-e,as -m m v— e e,asJ

X = - Vln(T(0) ). (5-67)

-q e,as

Equation (5-66) is the entropy balance equation. The first term

on the right-hand side is the entropy production, wich will be

shown to be positive definite. When Tj+ <*> it is clear from

(5-64) that in the case without a magnetic field the correction

f . had the general form:— GA

£ '" T^T- [ A ("^ X + B<">x ' (Te ee eA

where w is now the new independent velocity variable:

(5-69)m c 2

T ,eA

The functions A and B are solutions of the following equations

(see also appendix C and next section):

-85-

= w 3 / 2, (5-7Oa)

JCB = w3 / 2(w - 5 /2) , (5-7Ob)

where «6 is a symmetric, negative-definite integral operator.

With the definition of the fluxes 1 , and q . and the infor--eA aeA

nation just given the relations between the fluxes and the

forces read:1 ( D = m n

(0)|D ( 1 )v- 2X + D(1)X 1-eA e eA ' Te-m T -q

(5-71)

-q'

V 2

where: D U ; = -D JAJÜAdw, D^X) = -TlJvZAdv, D =3 . .ee

- (0 )v'To

(5-72)

' g j " ~ > --J ~gj » "g

o o 3v il

i< 1 5 - i TAZRHW i ( 1 ) - ! 7 R 2 R ^ i - "V'eA "Te\ n = -A JA*Bdw, A = -A JBotBdw, A = —————-—U eo 6o 6 3v /2

ee

One can clearly see that D and A are positive-definite

and that the Onsager reciprocity relations hold:m n . 1/ — v_, Art • \j~~to)

The entropy production rate is equal to:

( 0 ) . ( 1 )1 r e e A ( 1 ) ? r ( 0 / ( 1 ) TJ i ( 1 ) P I

Trv\ i ^ x™ + lm«n^A " T + JXm*X„ + ^ ^ i (5-74)(0) 2

m e e A T V2 ~m ~q q

eA Te Tewhich is proportional to:

» X 2 « X «X2 ) / 2 ^ 2 Z (5-75)

This expression is positive-definite if:

{JAZBdw} - /A^Adw/B/Bdw < 0, (5-76)O 0 0

which can be proved with the aid of the Schwartz Inequality.

In the following section the solution of the equations

(5-7Oa,b) will be decussed In detail.

-86-

V-4 The nonisotropic part of the electron distribution

In section 2 it was shown that the first order

contribution to the electron distribution function is

proportional to c only. It is also clear from the equations

that the inclusion of a magnetic field does not introduce extra

difficulties. In case of a zero magnetic field equation (5-48)

for f = f reads (see appendix D):

wjw °°j(-|x5/2 - ̂ x3/2)f(x)dx +(|w5/2 - |w3/2)/f(x)dx + 2w3/2f(w)0 W

|^(2wF(w)|£) - v(w)f(w) = w3/2b(w), (5-77)

/* , w 3 / 2 ,where: v(w) = -r— [v . + Jexp(w). (5-7S)

ee " T ( 1 ) ( W ) V T

The function F(w) was defined in (4-108), see also appendix D.

The problem is now reduced to solving euation (5-77), or:

2f = (J5 - v)f = w3/2b(w), (5-79)

where X is the part of X coming from the e-e collision term.

The other collisions are present in the function v(w). One can

easily verify that <£ and X are symmetric operators. When «C is

differentiated once a pure differential-equation of the Sturra-

Liouville type is obtained:

" W | f =-|j(ge"W), (5-80a)

(5-80b)

In the same way an operator J> can be obtained from

£ "W f | (5-81a)

v(w) )e"2w |f j +

+ e~W -|j(v(w)e"W)g(w), (5-81b)

Inspection of this operator shows that the last term of it

vanishes when the plasma is fully ionized, since then v(w)<* e .

-87-

For that case, which Is referred to as the Spitzer problem, the

operator 3) Is defined as followed:

J£] - [4F(w) + -^ eW]e-2wh(w), (5-82a)

J^h] =£&, h - Jf , (5-82b)

where t, = n.Z^/n Is the Ionic charge number,i i e

The accepted method to solve (5-77) is through an expansion

into a finite number of orthogonal Sonine-(Laguerre-)

polynomials11, which gives a set of linear algebraic equations

for the unknown coefficients in the expansion. This method

essentially is the Galerkin method, which can, of course also

be applied to the equations in differential form. There are,

however, some difficulties. It appears that the operator 3> is

not symmetric in all cases. By means of partial integrations

the following relation is obtained, valid for functions that

are bounded at w=0:

00 3f2 3fj -

J f ^ d w = [v(w){fl(w)_ - f2(w)— \]^Q + Jf/fjdw. (5-83)f2(w)0 0

From equations (5-70a,b) it appears that two source terms are

relevant: b=l and b=w-5/2. This makes the use of Sonine

polynomials of order 3/2 obvious, if one examines some

properties of these functions:

s(0) . ! s(l) = 5/2 - „ S( n ) = V r(n+5/2)(-w)k

b3/2 L' b3/2 D / Z W' b3/2 k^ o (n-k)!k!r(k+5/2) '

^ 3 ; 2 ^ 6nm. (5-84)

o

With the expressions for uC given in (5-77) it is possible to

calculate the coefficients:oo

X = /wpe~w £(wqe~w)dw. (5-85)q o

This is done in appendix D. The approach is some what different

from that in the litterature: the calculations presented in

-88-

appendix D are valid for arbitrary values of p and q, whereas

in the other calculations p and q are restricted to integer

values2/7/11. In terms of the operator «& the problem stated in

(5-79) reads:

jög = e~WJ^(w3/2b(w)). (5-86)

By means of partial integrations one can show that if gj and g2

are solutions of (5-86) with corresponding source terras bj and

b 2 and if f^ and f2 are the related solutions of (5-77) the

following identity holds:

00 CO

ƒfj2f2dw = /g1»g2dw, (5-87)o o

which directly gives the transport coefficients in (5-72) in

terms of the solutions of (5-86).

The matrix elements for the operator 3> are defined as:CO

6 = fwpA(wq)dw, (5-88)pq J

o

which are also given in appendix D. The calculation of these

coefficients is easier than for X ; they are also valid forpq

non-integer p and q. There still is one little problem: the

matrix is not symmetric for every set of functions, according

to equation (5-83). If p and q are natural numbers there is no

problem and 3> is symmetric. If p and q are non-negative

integers there is only one pair (p,q) for which the symmetry

relation does not hold:

*01 - A + ?10 = " W + k J(w-Dw2e~WQ(w)dw], (5-89)J01 " 4

" o

where Q(w) is related to the e-a collision cross section and is

defined in (4-106). In (5-89) it is also assumed that

lim W 3 / 2 T , 7 > ( W ) = 0. The parameter B is defined as:

6 = ™ee'2 = -^~ — = -^~ ̂ —^- {—J£-) , (5-90)

where r is the Landau length.

-89-

In the Galerkin method the solution of, say, equation (5-86) is

approximated by a linear combination of a finite number of so-

called co-ordinate functions <j> (w):

N

8M(w) = I a • (w) (5-91)n=0 n n

All these functions d> satisfy the boundary conditions and then

constants a are then fixed by the requirements:00

o ' w

If the functions to be chosen are <J> = w , the equations (5-92)

take the form:

N °°

£ \ n a n = /e"W|^[w3/2b(w)]<t,k(w)dw, (5-93)n=0 o

so that the matrix of the equations is not completely

symmetric. It can, however, be made symmetric if the first

equation (k=0) is replaced by:

00

/{o?fN - w3/2b(w)}e~Wdw = 0, (5-94)o

where X and f are related to J> and gN according to (5-81)—w

respectively. The function e is the solution of the homo-

geneous equation f=0. Therefore:CO CO COt —W "^ t —W f —W 3 t —W *\Je ££ dw = - Jv(w)e fN

dw = -Jv(w)e ~%~ie 8„Jd w- (5-95)o o o

Integrating by parts and using: e —[v(w)e J = ó61

one obtains:N f^ N

E a n / * n * l d w = ( 4 + ^00^a0 + Io " "* n=0 o n=0/e~W ̂ fNdw = ̂ Ia 0 + f an /«/id» = (^ + \QUQ + f . «

n=0

(5-96)

which shows that indeed the matrix is symmetric now. The system

of equations is not inconsistent, because the relation that

should be valid if (5-96) and the k=0-equation of (5-93) both

-90-

hold reads:

£N(°>

This relation, however, follows directly from the integro-

differentia! equation (5-77) for the exact solution. Thus it

may be expected that (5-97) is approximately satisfied with an

accuracy increasing with N.

Next an example is given: the calculation of the electrical

conductivity in the Spitzer limit with the aid of the operators *

jt> . In terms of this operator the problem then reads:

»ap = —(F(w)e"W - -J), p(w) = — h(w), (5-98)

where the right-hand side results from integration of equation

(5-86) with b=l; see also (4-109). The constant of integration

is chosen such that if w + « the right-hand side of (5-98)

becomes zero. On the basis of the general relation for the

diffusion velocity u . the first order electrical conductivity

is equal to:

, , 6itm v3 e2/2if

0 ( D = K e T e ° = 3j Kco (5-99)

2 ee2lnA e

where a . is the Lorentz conductivity of a fully ionized

plasma, i.e. taking only electron-ion collisions into account.

The constant K is related to the solution of (5-98) as follows:

O3 CO ^

K = -i- - — Jp*Spdw = i + J w V w /p(w')dw'dw, (5-100)

fy o o o

which again shows that the conductivity is always positive as

the operator Sb is negative-definite. The exact value of K has

been calculated numerically by Spitzer and Ha'rra1 and is equal

to 1.975 if c=l.

An approximation with polynomials can be made as follows:

NP(w) " P™ " I « »n« (5-101)

N n=0 n

If N is not too large there is no need for Sonine polynomials.

-91-

For N=l a system of two equations for aQ and

leading to:

K = I

results,

(5-l02)C 64<;2 + 244G/T + 288

To obtain the same result with the operator X, which has been

done by Landshof2 and Kaneko^, one has to solve three equations

for three unknowns. The numerical values of K for higher N

fully agree with their results. Substantial improvements,

however, can be obtained if non-integer powers of w are

admitted as co-ordinate functions. If p is approximated by:

p = pN

NI

n=0a wn

n/2 (5-103)

the result for N=l is even better than the fourth approximation

of Landshof. If N=2 the result cannot be distinghuished from

the exact Spitzer and Harm result, see table (5-1). If N=l the

result for K with approximation (5-103) becomes:

135ÏÏ- 32TT

256(

K = 17TT)

(15 4 2)ir - (2

(5-104)

N

1

2

3

Landshof2

1.9320

1.9498

1.9616

App.(5-101)

1

1

1

.9498

9616

9657

App.(5-103)

1

1

1

.9620

9751

9757

table (5-1):

values of K

for 5-1.

Near the origin the solution of (5-98) can be represented by a

Taylor series in powers of w 1 / 2 which could be an explanation

for the good results obtained with approximation (5-103).

-92-

This section is concluded with an examination of the limit of

very small degree of ionlzation. Equation (5-79) can be written

as follows (see also (4-106)):

^ w2Q(w)eWf(w) = - w3/2b(w) + (JS- £peW)f(w). (5-105)

If the degree of ionization is small 6 is a small parameter.

The solution of (5-105) may then be sought in the form of an

expansion in the parameter $. One then finds:00

f(w) = I f (w)Bn, (5-106)n=0 n

where: f. = - *£ M^£l , f = - j C L ^ e » - 4 - £}f . n > l ./w Q(w)/w " W2Q(w) /IT " l

(5-107)

It is readily verified that the first two terms of (5-106) are

equal to the first order contribution plus the multiple

collision parts of the second order contribution of the

function f . in case of a WIG. Thus the connection with the

weakly ionized gas theory has been verified.

-93-

References

1. L.Spltzer and R.Harm, Phys.Rev. 89(1953)977.

2. R.Landshof, Phys.Rev. ̂ É/ 1^ 9) 9 0 4» 82(1951)442.

3. S.Kaneko, J.Phys.Soc.Japan 15(1960)1685, 17(1962)390.

4. R.S.Devoto, Phys.of Fluids £(1966)1230, 22.(1967)354,2105.

5. W.L.Nigham, Phys.of Fluids 12(1969)162.

6. CH.Kruger.M.Mitchner and U.Daybelge, AIM J. £(1968)1712.

7. C.H.Kruger and M.Mitchner, Phys.of Fluids 10(1967)1953.

8. R.M.Chmieleski and J.H.Ferziger, Phys.of Fluids

10(1967)364,2520.

9. L.C.Johnson, Phys.of Fluids 10(1967)1080.

10. J.H.Ferziger and H.G.Kaper: "The mathematical theory of

transport processes in gases",

North Holland Publ. Comp. 1972.

11. M.Mitchner and C.H.Kruger: "Partially ionized gases",

J.Wiley, 1973.

-94-

VI NUMERICAL RESULTS

In this chapter the results of chapters IV and V are

applied to several practical situations. The shape of the

isotroptc correction is computed numerically for different

electron-atom cross sections. These are the hard spheres inter-

action model and the cross sections for neon and argon accord-

ing to experimental data obtained from litterature. The values

of the 36 basic coefficients k. ., which appear in the

expressions for the electron transport coefficients are given

for these cross sections. For other cross sections than the

constant hard spheres cross section these coefficients are

functions of the electron temperature.

Transport coefficients are calculated in several special cases

and are compared with results obtained by means of mixture

rules and with experimental results. When comparison with

experiment is made one has to bear in mind that not all

processes and effects have been taken into account such as

Inelastic collisions and impurities. On the other hand experi-

mental data suffer from rather large inaccuracies. These are

due to Several causes such as the lack of thermal equilibrium

and the presence of impurities.

Results obtained with the equations of the strongly ionized gas

(SIG) are also given and are included in some of the figures.

The better convergence with other functions than polynomials,

as shown already in chapter V for a fully ionized plasma, is

also observed in plasmas of a much lower degree of ionization.

In all calculations mentioned above it appeared that the cross

section of argon presents some difficulties, following from the

fact that it possesses a so-called Ramsauer minimum In the

energy range considered.

-95-

VI-1 The isotropic correction

In chapter IV the general solution for the first order

isotropic correction in a weakly ionized gas was given in

equation (4-113). There are six different functions J so that

there are in fact six isotropic corrections. See expressions

(4-107) and (4-126). In the numerical procedures the following

integration Is actually performed:

w G,(w')(6-1)

The solution of the homogeneous equation is then added after

the constants Cj and C. have been fixed by the requirements

(4-114). The isotropic corrections are given in figures (6-1)

to (6-3) for the cross sections of the hard spheres model

(hereafter denoted by HSM), and of neon and argon. The cross

sections for neon and argon were taken from references 1 and 2

respectively. The different isotropic corrections are numbered

according to the indices of the function G. ; see (4-107).

The reference cross section QQ has been chosen 10~20 mZ^ SQ

that the diraensionless functions Q(w) and hence the isotropic

functions are uniquely determined.

Characteristic for all isotropic correction functions is the

rather large peak near w=0 and the occurrence of two positive

zeros. The resemblance of the functions for neon and for the

HSM possibly implies that the HSM is not a bad approximation

for neon. The functions for argon have the same shape except

for the last two, and the magnitudes are larger than for the

other cross sections. This must be a consequence of the

Ramsauer minimum, which is absent in the neon cross section.

The coefficients k. . are also computed for these different

types of cross sections. See equations (4-125)-(4-127) for the

definition of these coefficients. They are the basic coeffi-

cients for the contributions of the isotropic correction to all

transport coefficients. Except for the HSM these coefficients

-96-

are functions of the electron temperature. In table (6-1) these

coefficients are give for three different cross sections. The

calculations were performed with a possible error of about one

percent, which is good enough when compared to the accuracy

with which the cross section data have been determined. The

constants for argon are significantly larger in absolute value

than for the other cross sections. Again this is due to the

Rarasauer minimum. This may invalidate the ordering and hence

severely restrict the applicability of the results.

The coefficients in expressions (4-132) to (4-135) for the

electron transport coefficients are algebraic functions of the

coefficients k .. Table (6-2) gives the values for the HSM,

while the results for neon are plotted as functions of the

electron temperature in figure (6-4). The temperature scale is

thereby chosen such that an atmospheric plasma in thermal

equilibrium in this temperature range is weakly ionized.

The effect of the isotropic correction can be demonstrated by

adding the zeroth order Maxwellian. This has been done In

figures (6-5) and (6-6) for an atmospheric argon plasma. The

other cross sections give similar results, see figures (6-1) to

(6-3). Figure (6-5) shows the influence of an electric field on

the isotropic electron distribution function and figure (6-6)

shows a similar effect due to a temperature difference between

electrons and heavy particles. From the source term (4-105) for

the equation of the isotropic correction it appears that

isotropic correction for a homogeneous plasma increases with

the square of the electric field and is proportional to the

temperature difference. The direction of the effect is the same

if the electrons have a higher temperature than the heavy

particles, as can be seen from figures (6-5) and (6-6).

When gradients are present the isotropic corrections numbered 3

to 6 are needed. For the special case of Maxwell interaction

between electrons and atoms there is an isotropic correction

only if a temperature gradient is present. See equation (4-115)

which gives the source term in that case. Therefore this model

-97-

seems to be less suited for a description of the electron-atom

interaction than the hard spheres model.

j-

0.40

-0.87

-0.87

-6-41

-0.10

0.93

-1.26

2.35

2.35

15.2

0.30

-2.50

0.81

-1.75

-1.75

-12.8

-0.21

1.85

-3.43

8.02

8.02

63.9

0.93

-8.49

3.96

-7.63

-7.63

-50.9

-0.96

8.11

-9.29

21.9

21.9

176

2.52

-23.1

Table (6-la): k. . constants for hard spheres model (HSM)

.078

-0.30

-.046

-0.57

-.0026

.046

-2.74

7.78

1.14

12-4

.075

-1.19

0

-0

-0

-1

-.

0

.14

.77

.12

.63

0058

.12

-1

6

0

.25

.15

.95

12.0

.

-0

046

.95

6.

-20

-2.

-32

-0.

3.

99

.2

98

.6

19

10

-3.

17

2.

34

0.

-2.

59

.9

77

.7

13

75

Table (6-1b): k. . constants for neon at T = 5000K.ij e

-4.

21

-4.

14

-3.

7.

59

.9

30

.2

16

16

-101

113

-54.4

148

-7.16

23.2

-45

56

-25

69

-5.

13

.5

.0

.6

.9

05

.7

16

-56

12

-39

7.

-16

.1

.2

.5

.9

16

.6

208

-226

110

-301

14.3

-44.3

102

-134

57.0

-158

11.1

-30.5

Table (6-lc):k. , constants for argon, T = 5000K, data Milloy2

-93-

3.32

11.2

-0.79

1.00

-2.30

4.39

-78.1

112

-52.0

133

1.01

11.9

-35.4

64.9

-27.3

67.5

-2.15

11.0

7.48

-43.7

10.9

-26.1

4.39

-10.9

160

-215

103

-264

-4.69

-18.7

87.6

-147

63.5

-159

2.50

-21.1

Table (6-ld): k . constants for argon, T = 5000K,

data Frost and Phelps3.

k 1 2 =

k52 =k 2 2 =

K ~

K2 "k 24 =

S "

-1.26

0.30

2.35

149

31.1

8.02

14.1

k 1 4

K3k 5 1 +

K5

k 5 1k n

= -3.43

= -13.6

= 0.93

= -2.61

= 0.38

= -0.10

= 0.40

k 23 =

<? =

k 2 1 =

k 2 5 =

^25 =

Y3 5 "

-1.75

5.02

-0.87

-7.63

-62.5

-7.93

7.63

Table (6-2): Some coefficients for the HSM appearing in

equations (4-132)-(4-135).

-99-

5

-1 -

-20 1 2 3 4 5 6 7 8

-40 1 2 3 4 5 6 7 8

Fig.(6-la,b,c) Isotropic correction functions for the HSM.

-100-

0 1 2 3 4 5 6 7 8

0 1 2 . 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

Fig.(6-ld,e,f) Isotropic correction functions for the HSM.

-101-

1

n

i

2

30

//

JJ1

1 1 1

I I I

2 3 4W

1 a

NEONTe=5000K .

| |

5 6 7

3

M-l

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

Fig.(6-2a,b,c) Isotroplc correction functions for neon1,

at T = 5000K.e

-102-

1 2 3 4 5 6 7

. 1

* O

- . 10 1 2 3 4 5 6 7

3Is

-1

-2

1 1 1

J1 1 f

1

• ^ - ^

1

1 ' f

NEON '

V 5000K

0 1 2 3 4 5 6 7UI

Fig.(6-2d,e,f) Isotropic correction functions for neon1,

a t T = 5000K.e

-103-

1 2 3 4 5

Fig.(6-3a,b,c) Isotropic correction functions for argon,

at T = 5000K.e

-104-

0 1 2 3 4 5 6 7

Fig.(6-3d,e,f) Isotropic correction functions for argon,

at T = 5000K.e

-105-

m

• y

•

•

/

1 1 !

11 -

4000 5000 6000 7000TEMP. (KELVIN)

10

8

6

4

Z

-2

^ —

k

^ — — ;

. k u •

SI

1 , 1 . "

4000 5000 6000 7000TEHP. (KELVIN)

4000 5000 6000 7000TEMP. (KELVIN)

60

50

40

30

20

10

—

'- - y

- &s1 1 . . _

• i

i f•

• t

i

s

4000 5000 6000 7000TEHP. (KEIVIN)

4000 SOOO 6000 7000TEHP. (KELVIN)

-1-44000 5000 6000 7000

TEMP. (KELVIN)

Fig.(6-4) Several coefficients appearing in the transport

coefficients for neon1; see equations (4-132)-(4-135)

-106-

0 1 2 3 4 S 6

Fig.(6-5a,b) Effect of an electric field on the isotropic part

of the electron distribution function( ) in

the case of argon2:

p = latm., n = 1.3 1018nT3,T = T = 5000K.

M: zeroth order Maxwellian without isotropic correction.

-107-

Fig.(6-6) Effect of a temperature difference on the isotropic

part of the electron distribution function in case of

argon2: p = 1 atm., n = 3.4 1017nT3,

T = 4500K, T = 1000K.e ' a

M: zeroth order Maxwellian without isotropic correction.

-108-

VI-2 Electron transport coefficients

In this section the electron transport coefficients in

weakly ionized gases (WIG) are calculated from the expressions

(4-132) to (4-138). Some results of the strongly ionized gas

(SIG) of chapter V are also given. The first order parts

contain the coefficients S0,R0 and LQ which are functions of

the electron temperature except for the case of the HSM. If the

electron-atom interaction potential is assumed to vary with

some power of the interaction distance, the collision cross

section is proportional to a power of the relative velocity. If

this model is adopted one has:

T ( 1 ) ( » ) = V 'n , (6-2)

so that:

-, . -(nri-l)/2 T - 1 . (6-3)Q(w) = q W v ' , q = —pr , T = V '

m T (2v ) m / 2 n /2 v„ Qnm Te a Te O

The coefficients mentioned above are then easily calculated

giving the following results:

The hard spheres (HSM) corresponds to m=-l with q-^1.

The following coefficients can be calculated exactly for the

interaction model (6-2):

bee qra m/2,m/2 ' ee qm l m/2,ra/2+l

Lee " qm ^m/2+l,m/2+l 5Xm/2+l,ra/2+ 4 -1' ( 6 5 )

where the coefficients X are defined in (D-41) of appendix

D3. A tedious but straightforward calculation gives for the

HSM:

S = - [-Ü- + TT - T-lnC1-̂ ) 1*̂ » -0.2276e e 30^2 1 5 4

-109-

Re e

= " \r^l= -%~ r-lnd+^J/ÏT = 0.6436e e 60/2 iU ö

[ gln(l+/2)]v^ -1.8775- (6-6)e e 120/2 b

If more realistic cross sections, which are available in the

form of tables, are used these coefficients have to be calcula-

ted numerically. It turns out that the coefficients in (4-121)

for the multiple collisions are very sensitive to the precise

shape of the Rarasauer minimum of argon. This is demonstrated by

using two different cross sections, one from Milloy2 and one

obtained earlier by Frost and Phelps3. Figure (6-7) shows a

sketch of these cross sections and in figure (6-8) a plot is

given of the function:

SF(w) = ƒ -̂ Z{—~ }dx, (6-7)o Q(x)/x Q(x)/x

which is related to one of the coefficients in (4-121), namely

S = S17(«>). One can then see that the main contribution to the

integral S comes from the Ramsauer minimum. It is clear thatee

the sharper minimum in the cross section data of Milloy et al.

results in a much larger value of See

The electrical conductivity will now be calculated as a

function of the parameter 3» defined as:n , » rT 2

. - pr e inA L , , „.3 - veeT/2 = — - ^ J J - , (6-8)

a u

where r is the Landau length, see chapter II. In the presence

of an electric field and a temperature difference between the

electrons and heavy particles the electrical conductivity in a

uniform WIG, where 0 is of the order e, up to second order

reads:

a(1) + a(2) =

• (6-9)e a

-110-

From this expression one can infer that when g is either very

small or very large, singularities occur originating from the

fact that the ordering has a restricted region of validity.

Comparison is now made with three other calculations of the

electrical conductivity. Firstly the addition mixture rule

introduced by Lin et.al.1*, which is defined as follows:

where a = OQSQ is a result of electron-atorn collisions only

and a of electron-ion collisions only:ei

64/27e2v3

oel := - 2 J 2 . , (6-11)Z1e

2lnA

and Yp *s t n e well-known Spitzer factor: y„ = 0.582, so that

Y„a . is the electrical conductivity of a fully ionizedE ei

plasma5. Mixture rules proposed by Frostfa, use the lowest order

expressions for the transport coefficients in a WIG, but add to

the electron-atom collision frequency a modified electron-ion

collision frequency in order to obtain simple formulae for the

transport coefficients which might be reasonable approximations

for arbitrary degrees of ionization, from the weakly ionized

gas up to the fully ionized plasma. Care has been taken that

the expressions give the correct answer in the fully ionized

limit. In case of the electrical conductivity the Frost mixture

rule reads:

P t w*5 ^e dw

o* = oj £_222 . (6-12)o {w3/3Q(w)+0.952s}

The third way of calculating the electrical conductivity is

based on the equations of the SIG, see section 4 of chapter V.

For the case of a HSM the convergence is good, especially when

powers of half an odd integer are admitted as co-ordinate

functions. In table (6-3) two sets of co-ordinate functions are

compared with each other. One consists of the classical Sonine-

-111-

or Laguerre polynomials and the other one is a set of orthogo-

nal functions constructed by means of Gram Schmidt's method

from the following functions:

3/21, w, w

The function

(6-13)

is not permitted because it results in

infinitly large matrix elements 6 . In table (6-3) values of

-(A,J5A) appearing in equation (5-72) are tabulated for neon at

an electron temperature of 5000K, with an increasing number of

co-ordinate functions up to eight.

Number of

functions

2

3

4

5

6

7

8

Polynomials

1.3393

1.3842

1.4128

1.4264

1.4325

1.4354

1.4370

Functions

in (6-13)

1.3393

1.4069

1.4370

1.4401

1.4402

1.4402

1.4402

Table (6-3), values of -(A,J5A) for the HSM with 0=1.

When 3 * 0 all calculations of the electrical conductivity

except (6-9) converge to the first order part of expression

(6-9), because none of them takes any deviation from a

Maxwellian electron distribution into account.

Figure (6-9) gives the results of the calculations for the

HSM; i.e. when Q(w) = 1. To obtain clear pictures the conducti-

vity is normalized to QQSQ for low values of 3 and to the

Spltzer conductivity y„o , for the higher values. The relation

between these normalizations is the following:

qosoVel

3» (6-14)

-112-

One can see in figure (6-9a) that the electric field suppresses

the electrical conductivity below the common limit of the other

calculations. Figure (6-9b) shows the strongly ionized domain

where the addition rule gives much higher and the Frost mixture

rule gives lower values for the electrical conductivity than

the SIG calculations. Figure (6-10) shows similar results for

neon.

Calculations of the thermal heat conductivity are given in the

next two figures. Figure (6-11) shows the results for the HSM

and (6-12) for neon. As can be seen from these figures, the

Frost mixture rule gives rather good results in the SIG domain.

When gradients are weak the expression for the thermal heat

conductivity up to second order reads:

where <k = k26 - kTk25 - %k2k.

For large 3 only the Frost mixture rule and the SIG results are

shown in figures (6-11) and (6-12), normalized to the Spitzer

value. For small values of g the normalization is done with

respect to (L - k R0)XQ, i.e the first order contribution.

When T /T = 0.9 and T = 5000K, the thermal heat conductivitya e e

at lower values is higher than the results of the Frost mixture

rule and The SIG in the case of neon, see fig (6-12). At higher

electron temperatures the effect changes sign because < does,

see fig (6-4).

The cross section of argon leads to many difficulties, because

of the large values of the occurring coefficients. If the

fields and gradients are small enough, reliable results may be

obtained for low degrees of ionization.

-113-

Fig.(6-7) Plot of the data for the electron-argon cross section

for momentum transfer as obtained by Milloy2( )

and by Frost and Phelps3(- - - - ) .

ÜJ

v

Fig.(6-8) Plot of the function S (w), see (6-7) for the cross

section data of Milloy(M) and of Frost and Phelps(FP)

at T = 5000K.e

-114-

1.1

to 10

Fig.(6-9a) Electrical conductivity normalized to the zeroth

order value a S for the HSM.o o

1

• 9

• 8

.7

_2L"6

V».4

.3

.2

.1

0

HSM.....I

10° 10 JO*(3

10'

Fig.(6-9b) Electrical conductivity normalized to the value of

the fully ionized plasma for the HSM.

B = 2 v T , AR: Addition mixture rule, FR: Frost mixture rule,ee

-115-

1.1

1

9 •

•6 I-

•5

•10"

/ NEON• Te*S00OK

" e E = 0.01

WI6\

-..AR

\

-J • x

10 10

o-t 10°

Fig.(6-10a) Electrical conductivity normalized to the zeroth

order value for neon at T = 5000K.e

10 10

Fig.(6-10b) Electrical conductivity normalized to the value of

the fully ionized plasma for neon at T » 5000K.\ e

f$ = 2 v T , AR: Addition mixture rule, FR: Frost mixture rule.

-116-

10 10

Fig.(ó-lla) Thermal heat conductivity normalized to the zeroth

order value for the HSM.

10

Fig.(6-llb) Thermal heat conductivity normalized to the value

of the fully ionized plasma for the HSM.

8 = 2 v T , FR: Frost mixture rule.ee '

-117-

1-2

Fig.(6-12a) Thermal heat conductivity normalized to the zeroth

order value for neon at T = 5000K.e

10

Fig.(6-12b) Thermal heat conductivity normalized to the value

of the fully ionized plasma for neon at T = 5000K.

6 = 2 v T , FR: Frost mixture rule,ee

-118-

VI-3 Electrical conductivity of cesium seeded argon plasma

When for a seeded plasma calculations are compared with

experimental results there is the advantage that for these low

temperature plasmas the experimental conditions are well

defined. On the other hand, however, the low temperature and

the relatively high degree of ionization tend to give high

values of (3 outside the scope of the present theory of the UIG.

It therefore appears that the experimental values are not in

the region where the isotropic correction influences the

transport coefficients significantly. Calculations have been

performed for a cesium seeded argon plasma of atmospheric

pressure. Two different cesium-cross sections have been used

together with the argon cross section of Milloy2: one has been

obtained by Postma7 and the other one by Stefanov8. Figure

(6-13) shows these rather different cross section data.

In figure (6-14) the experimental points are from Harris9,

which contain a possible error of 30%. The results with

Stefanov1s cress section are In better agreement with the

experimental points simply because he used Frost's mixture rule

and the data of Harris to fit a curve for the cross scetion of

cesium. Postma, however, used electron drift-velocity measure-

ments and numerical integrations of the electron Boltzraann

equation to obtain his curve. The results of Postma are not too

far away from the experimental points. The fact that the

experimental results are larger than the theoretical ones for

higher cesium-pressures has also been observed by Kruger et

al.10. It might thus appear that curve fitting of cross

sections by means of experimental data is rather inaccurate, if

possible experimental errors are high. The difference between

the Frost mixture rule and the present work for Stefanov's

cross section is due to the fact that the minimum in his data

lies at the same energy value as the Rarasauer minimum of argon

and thus reinforces the Influence of the latter.

-119-

f * 7 •iitcirr

Fig.(6-13) electron-cesium momentum-

transfer cross section

data as obtained by

Postma7(P) and by Stefanov8(S).

1300 1400 1S00 ISOO 1700 1600TEHP. (KELVIN)

10

0.1 torr Cs

Stttanov

1300 14D0 1S00 I tOO 1700 1100TEn*. (KELVIN)

1300 1400 1500 1600 1700 1800TEflP. (KELVIN)

10

£ 10

10'

10

1 torr Cs

Stefanov

1300 1400 isoo isoo 1700 leoo

Fig.(6-14) Electrical conductivity of cesium seeded argon

plasma as a function of the electron temperature.

e-Ar cross section data of Milloy2

e-Cs cross section data of Postma7 and of Stefanov0.

argon pressure is 1 atm.

-120-

References

1. A.G.Robertson, J.Phys.B 5^(1972)648.

2. H.B.Milloy et al, Austr.J.Phys. 30(1977)61.

3. L.S.Frost and A.V.Phelps, Phys.Rev. 136(1964)A1538.

4. S.C.Lin,E.L.Resler and A.Kantrowitz,

J.Appl.Phys. 26(1955)95.

5. L.Spitzer and R.H'arm, Phys.Rev. 89(1953)997.

6. L.S.Frost, J.Appl.Phys. 32(1961)2029.

7. A.J.Postnia, Physica 43(1969)465.

8. B.Stefanov, Phys.Rev. A22_( 1980)427.

9. L.P.Harris, J.Appl.Phys. 34(1963)2958.

10. CH.Kruger.M.Mitchner and U.Daybelge, AIAA J. £(1968)1712,

-121-

VII SUMMARY AND CONCLUSIONS

The work presented in this thesis shows that a perturba-

tion expansion in the framework of the multiple time scale

formalism is well suited to attack the complicated set of

kinetic equations describing transport phenomena in a partially

ionized gas.

The equations are limited to elastic collision processes only.

The aim of the present work is to describe transport phenomena

at thermal energies and to calculate transport coefficients. \s

partially ionized gases are in general low temperature plasmas

of thermal energies well below the first excitation level the

restriction above is not unrealistic.

In chapter II the basic considerations are given. For the

description of the Coulomb collisions the Landau collision

integral Is used, which assumes a static screening of the

charged particles in the collision process. The other

collisions will be described by the Boltzraann collision

integral, valid for short range interaction potentials and

assuming only binary interactions of the collision partners.

Diverse parameters in the problem are related to the principal

small parameter: the square root of the electron-atom mass

ratio. Among these are the electric field and the Knudsen

number but also the degree of ionization. The latter is used to

make a division of partially ionized gases into four categories

from very weakly ionized to strongly ionized.

In chapter III the necessity of a first order isotropic correc-

tion in a very weakly ionized gas, when only electron-atom

collisions are taken into account, is proved for a situation

with the same ordering of parameters as that of Bernstein. He,

however, incorrectly assumed that a possible correction could

be absorbed into the zeroth order electron distribution

function. When the plasma is homogeneous the equation for the

zeroth order electron distribution function describes the

relaxation towards the Davydov distribution on the timescale

-122-

for energy relaxation between electron» and atoms. This equa-

tion can be completely solved by means of separation of

variables and subsequent solution of an eigenvalue problem.

The case of a very weakly ionized gas, with Coulomb collisions

included, is complicated because of the nonlinearity of the

equation for the zeroth order electron distribution function.

This equation describes the competition between the tendencies

to establish a Davydov or a Maxwell distribution function. All

corrections to this function are functionals of it so that

solving that equation is essential for that region. The

equation has been brought into a rather simple form, which may

lead to possible analytic and-or numerical solutions. An analy-

tic solution for the tail of the electron distribution function

has been obtained.

Also for a weakly ionized gas, as studied in chapter IV, an

isotropic correction has to be introduced. The integro-

differential equation for it is solved analytically. It appears

that for a given electron-atom cross section there are in fact

six different isotropic correction functions. Six moments of

each of these functions are needed to evaluate the corrections

on the transport coefficients. Thus there are 36 coefficients,

which, apart from the hard spheres interaction model, are

functions of the electron temperature. New transport phenomena

are found which depend nonlinearly on the gradients and forces

or involve second order derivatives.

The second order corrections to the transport coefficients can

be devided into two groups: one of them consists of nonlinear

contributions from the isotropic correction, the other

encompasses the effects of multiple collisions. The latter give

linear relations between fluxes and forces and thus obey

Onsagers' relations.

Chapter V deals with the strongly ionized domain. The equation

for the first order non-isotropic part of the electron distri-

bution function has been cast in the form of a fourth order

differential equation.

-123-

The calculation of the coefficients needed for the Galerkin

method happens to be easier in this formulation. Th- Spitzer

equation for the non-isotropic part of the electron distribu-

tion function reduces to a differential equation of even second

order, which provides a more convenient basis for the calcula-

tion of the transport coefficients than the equation of Spitzer

and Harm. When powers of half an odd integer are admitted as

co-ordinate functions the convergence of the Galerkin method

becomes much better.

The results of the numerical calculations in realistic cases

are collected in chapter VI. The domain in which the results of

the weakly ionized gas can be fruitfully applied strongly

depends on the energy dependence of the electron-atom cross

section for momentum transfer. Especially in the ca^e of argon

the results are so poor that the domain almost vanishes. This

is due to the well-known Ramsauer minimum.

Of the mixture rules that are tested, the addition rule gives

too high values for the transport coefficients while the Frost

mixture rule appears to be relatively reliable. The electrical

conductivity is also calculated for a cesium seeded argon

plasma and compared with experimental results. This is done for

two rather different experimentally obtained sets of data for

the cesium cross section, of which those obtained by Postma are

more reliable than those obtained by Stefanov.

At lower cesium pressures the agreement between theory and

experiment is satisfactory, while at higher pressures the

theoretical values are lower than the experimental ones. It

should be remembered that the experimental conditions are at

the border of the range of validity of the present theory for

the weakly ionized gas.

-124-

APPENDIX A

Expansion of the electron-heavy particle collision Integrals

A-l Electron-atom collisions

The Boltzraann collision term describing elasic collisions

between particles of species s and t in a dilute gas is usually

given in the following form:

SC» f ) = / d 3 g / b d b d < f > g { f ( v ' ) f ( v ' ) - f ( v ) f . ( v ) } , ( A - l )

t S C- u S C C

where b denotes the impact parameter and g is the relative

velocity: g = v - v. Primes indicate post collision variables,

which are defined by the relations:

ml

Conservation of momentum is guaranteed by these expressions.

Conservation of energy in the centre of mass reference frame

requires: g'=g- This can also be expressed by the relation:

I2 + 2g.* = 0. (A-3)

It is now possible to show the integrations to be performed

explicitly if the differential cross section o(g,x) *s defined

by:

J4H

bdbdij) = b — dxdifi = - a(g, x)sinxdXdli>> (A-4)

where x is tne scattering angle in the centre of mass system.

Figure A-l shows an encounter in that system. With the help of

relation (A-3) the integration over x a"d <f> is written as an

integration over the complete Jl-space. Expression (A-l) then

reads as follows:

J (f ,f ) = 2/d3g/d3JlI(g,Jl)6(A2 + 2g.*)xst s t - -

Xt f s<v - !TTm-)ft<2 + 8 + TlZr> - f Js t s t

-125-

<A-6)where: I(g,*) = o(g,X) = " ^ ||^| •

The 6-Dirac function takes care of relation (A-3).

The collision Integral J describing the collisions betweenea

electrons and atoms will be expanded in powers of the small

parameter e:

e = (m /m ) .v e aJ(A-7)

If the electron temperature is of the order of the atom

temperature the velocity variables in the thermal range scale

with e. A Taylor expansion then gives the following result:

ea ea

èa

ea

ea(A-8)

(A-8a)

/d3v v f (v Wdfla(v,x)2(n2 - I)-Vf(u) +3. a a. a V G

' j 3 v Z f <*><J"dnVJva<v>x)){f (u) - f (v)}, (A-8b)

^d3v av av af a(v a):/dn{Vv7v(va(v,X))(f e(u) - fg(v)) +

-126-

4V (vo(v,x))(I-nn)«V f (u) + 4va(v,x)(I-nn)(I-nn): V V f (u)} +

2m—-/d 3v f (v )Vm a a ~a va

- nn)-vva(v,x)f (u),— — e —

(A-8c)

In these expressions u Is the electron velocity after a

collision with an atom of infinite mass, so that u=v. The unit

bisector of the angle between u and v is n, see figure A-2.

'Jhat is left from the

integration over A-space

is an integration over

dfl = sinxdxdiji.

The introduction of the

unit vector n permits

the following notation:

u = (2nn-I)«v. (A-9)

A transformation in velocity space to a reference frame moving

with the hydrodynaraical velocity of the atoms makes the first

order term (A-8b) equal to zero. The expression can be simpli-

fied further if it is assumed that f is a Maxwellian. In that

case the third order contribution vanishes also:

Jea

J(0>ea

ea ea

/diJv(v,X)[f (u) - f (v)],

kT

(A-10)

(A-lOa)

7 f (u)v e -

n va(v,x)3.

(A-10b)where: v(v,x)

The collision operator permutes with the rotation operator in

velocity space. Therefore the spherical harmonics or equi-

valeutly the harmonic tensors of appendix C are eigenfunctions.

If f is expanded into harmonic tensors:

-127-

(A-ll)n=0 " c

and if this expansion is substituted into (A-lOa) the n-th term

reads:

nfe(v)«/dfiv(v,x)[<un> - <vn>]. (A-12)

Consider now the right-hand factor of this n-fold dot product.

After having performed the integration over ds} = sinxdxd<|> where

X is the angle between u and v, the result still is a harmonic

tensor of rank n, therefore:

gn ('

where g is as yet undetermined. By taking the n-fold inner

product of this expression with <v > one can show (see app. C)

tha t g Cx»*!") = P (cosx) so t h a t :n n

J ( f (v )*<v>) = f (v) »<v >/dS2v(v,x) JP (COSY) -1 }. (A-14)ea n— n — n— n— l n '

In general with expansion (A-ll) for some function f :

(0) v i riJ (f ) = - T T~ 1 (v) f (v)«<v> (A-15)ea e _. (1) n—e n —

with: T 1

( y ) = /dJMv,x)[l - Pn(cosx)].

Note that the term n=0 is absent: J is zero for an arbitrarye a (0)

isotropic function. In appendix B it is proved that J is

symmetric and negative-definite. Also an H-theorera will be

derived.(2)

The expression for J becomes also very simple if f isisotropic:

If f is a Maxwellian too, this expression becomes:e

Which shows that this term vanishes in thermal equilibrium.

-128-

A-2 Electron-ion collisions

The interaction between charged particles will be

described by the Landau collision integral; see also chapter II

and appendix D. The general form reads as follows:

2 1 —/f » f

o ) = C OV • / — . { — 7 - — V , }f (v)f o

a ' ?>' a6 v J ? lm v mn v' ' a - gg a p

— 7 - — V , }f ( v ) f o ( v ' ) d 3 v f

(A-17)where: g = v 1 - v and C i s a cons tan t defined by:

- - - ap

q l

SLÊ (A-13)

o a

The case in which the particles have equal masses will be

treated in appendix D. The electron-ion collision integral can

easily be expanded in powers of e by means of Taylor

expansions; the results up to second order are:

(0) l , ,v *•= v e;'ei m v *•= v e

e

j _ei ra

e

- _£ijVf (V>)d3vt.v .[(V V ) . v f 1 (A-20)ra - I - v L v= v eJ

C n 2 v CU > = _ e i 1 7 r _ ^ f > + e i_ j . , , f ( v . ) d 3 v » : 7 . [ ( v v V ) - V f 1e i m v ^ 3 eJ 2ra - - i - v •• v v = v e J

(A-21)where the tensor V is defined as follows:

v 2 l - vvV := = 7 7 v. (A-22)= 2 v v

Some properties of V, which are easily verified are:

V«v - 0, (A-23a)

2vV .v = - -^-, (A-23b)v " V3

v-7 V = -V, (A-23c)

-129-

v ( V V V) = -2v V , (A-23d)- v v= v=

A V = 7 •? V = — c <w>, (A-23e)

v= v v= vD — '

= f-(v2 |-) + v37 «(V-V ). (A-23f)

With these properties It can be shown that:

co C n.

<O> ei i

(f ) I n(n+l) ei i f ( v ) ' ^ 1 ^ , (A-24)

ei e L, a n-e n -n=l m v*

e

and if f is isotropic:C 2v

m

i_I2|_[I|l). (A.26)e

The second term on the right-hand side of (A-26) clearly

vanishes if f is also isotropic. If the electron and ion

distribution functions are Maxwellians with hydrodynamical

velocity equal to w and with different temperatures one finds

that the first contribution to J is given by (A-26) and

reads:

e e

See also equation (A-16a) which is of the same type.

-130-

A-3 Moments of the electron-heavy particle collision Integral

All collision integrals considered are elastic and thus

preserve the number of particles:

/J (f ,f )d3v = 0, s,t = i,e,a. (A-28)st s t

In general the moment with some function <j> of v reads:

/Jst(fs,ft)<t.(v)d3v. (A-29)

If J is the Boltzmann collision integral of (A-5) and thes c

following transformations:m %

I + -I, g * g + *, Y. + Y. " ~-' (A-30)o

are applied in this order one obtains for the moment (A-29):

/d3V(j)(v)J (f ,f ) = - 2/d32d3vd3g6(£2 + 2g-A)Kg, £)*- st s t - -

m i" [ • (v - r ^ 1 ) ~ • (v)J f ( v ) f . ( v + g ) , (A-31)

o

which indeed is zero if $=1. If $=m v equation (A-31) becomes:

2m m2/d3vm vJ = 2_£/,i3va3gd3u6(Jl2 + 2g-l)I(g,l)f (v)f (v + g)0

(A-32)Firstly the integration over £-space is performed in spherical

co-ordinates with g as polar axis:

fdH A6(A2 + 2*.g)I(g,A) = - -— T-Y , (A-33)n T ( r 8 J

where T-1. is defined as in (A-15). Thus:

( A" 3 4 )

If s=e,t=a this term can be expanded in powers of e. It is then

convenient to perform an integration over v'= v + g instead of

one over g. The# result of the expansion for this case is:

- 1 3 1 -

m yf (v) m m v kT/d3vm vJ = - J 6 e , \ d3v + e 2 - i r f e " _ a v A (—£___)+

e— ea J T . 1 . ( V ) m T , . . ( V ) 2 — v k i , . , ( v ) J

- kT V [ ^ -T- l l f (v)d3v +&(t1*). (A-35)a V t ( l ) W e ~

Next iji(v) = 5jm v2 i s inserted into (A-31):

mm m X.2

Am v2J d3v = - 2 - VS SC Ïs st m o v m - -' i

o tx f (v)f (v + g). (A-36)

S ™ C "™ *•*

With the aid of (A-33) and s similar integral:

2

t (1)

the moment integral with ̂ m v2 can be written as follows:

smm m g 2 f (v)f (v + g)

Am v2J d3v = -=-=vfd3vd3g(—=—+ g - v } - ^ — " . . — . (A-38)s st mQ m o -T n tT (1 )(g)

If again s=e and t=a the following expansion in powers of c is

obtained by means of Taylor expansions of the integrand:

ra 3kT - ra v 2

Am v 2 J d 3 v = e 2 —ƒ f r—r 1- kT v-r—f ) If ( v ) d 3 v + Oiz*)s ea m J L T ^ . - C V ) a 3 V V T . 1 ^ ; J e -

(A-39)

Finally the results are given of the expansions of the moments

of the electron-ion collision integrals:

vfm vJ .d 3 v = - 2C , n . f — f (v )d 3 v +

e- a i e i xJ 3 e -

3<v2> 2m v- e2C . / v ' f . ( v ' ) d 3 v ' « f f ( v ) d 3 v - e2—^C .n,J—f ( v ) d 3 v +

ei - i - v 5 e - m1 ei i Jv 3 e -

15<v3>- e2 /y 'v 'f i(v l)d3v< : / —fg(v)d3v + #(e3) . (A-40)

vAmev

2J fd3v = -e2Cel/v

1f1(v')d3v t ' f—i (v)d3v +

-132-

2C m <v2>_ e 2 — Ê l ^ / i f (V)d3v -e26C . / v 1 v ' f , ( v l ) d 3 v 1 : f ~ — f (v)d3vm. J v e - e l 1 - J c e -1 v 3

Some caution must be taken with integrals like:

<V2>ƒ——f(v)d3v. (A-42)v5

At first glance this integral would be zero If f is Isotropic.

But this is only true if f(0)=0. The result of the integration

must be an isotropic tensor of rank two (see also app.C):

<V2>ƒ f(v)d3v = -j/VvVv(i)f(v)d

3v = A(v)I. (A-43)v5

Contraction on both sides gives:

3A = j/Av(^)f(v)d3v = - YL/5(v)f(v)d3v = - *ff!;(0). (A-44)

so that: ƒ——f(v)d;iv = - ^v I. (A-45)

A generalization of (A-43) is:

JV^[i)f(v)d3v = An(v)nI, (A-46)

where I is an isotropic tensor of rank n. For odd n there isn-

no such tensor so that these integrals are zero. This follows

also from the fact that the Integrand is an odd function of v.If n is even A can be determined by a contraction over all

n

indices and a subsequent calculation of the Integral as

follows:

|A"/2(|)f(v)d3v = - 41r/(AjJ/2"1ó(v)]f(v)d3v. (A-47)

If E can be differentiated a sufficient number of times this

integral can be calculated.

-133-

APPENDIX B

Some H-theorems and properties of collision integrals

B-l The zeroth order electron-atom collision operator

The zeroth order electron atom collision operator J has

ea

been derived in appendix A. In this section two properties and

an H-theorem will be derived. With tne aid of the 6-Dirac

function the collision integral .7 can be written as follows:J(0)(f) = n /d3£6(Jl2 + 2A.y)I(v,je){f(y - 1) - f(y)}. (B-l)

By multiplication with some function g and integration over the

entire velocity space the following inner product is obtained:

f),g) = n /d3vd3£6U2 + 2y««)I(v,JO{f(y-£) - f(y)}g(y).

(B-2)If the velocity transformations v •*• v - X. and I •*• -I are

performed in this order, (B-2) is found to be equal to:

(0)*• e a a — — '

x{f(v-A) - f(v)}jg(v) - g(y-A)} = (J^^g)^], (B-3)

so that J is a symmetric operator. From this expression oneGel / r\\

finds immediately that J is negative-definite:

(j^(f),f) < 0. (B-4)

Finally an H-theorera can easily be proved as follows. Suppose

that the following equation holds for f:

| | = JgaJ(f)- (B-5)

The quantity H is now defined as:

H = /fln(f)d3v. (B-6)

If equation (B-5) is multiplied by: 1+ ln(f),and integrated

over v-space, tha following inequality can be obtained:

-134-

3H tta f ( V >

= ^Jd3v43lS(l9 i 2 v « j ) I ( v A ) l n ( 3 ) { f ( v - A ) - f ( v ) J < O,•— = ^ • J d v 4 l S ( l i 2 ( )

(3-7)

because f is assumed to be positive everywhere. At the same

time H is bounded from below, so that when t •* °° the integral

in (B-7) is equal to zero and f has become an isotropic

function, i.e. depending only on |v|.

B-2 The zeroth order electron-ion collision integral

The above derivations suggest that the properties of J(0) e a

also hold for J , . They will be proven below. One has:

^ jpV^V v ve e (B-8)

which is symmetric because V is symmetric. If g=f the

integrand in the right-hand side of (B-8) takes the form:a2v2 -' (a«v)2

a-V«a = — — ? 0, (B-9)

v3

which shows that J , is a non-positive operator. Finally an H-

theorem is derived. Again H is the quantity defined in (B-6).

If in (B-8) g = 1+ ln(f) and f obeys:

it is easily demonstrated that:

-!•£ = - 1 el/i-V f«V«V fd3v < 0. (B-ll)3t m Jf v = v3t m Jf v

e

-135-

B-3 H-theorems for the ion distribution function

In this section an H-theorem will be derived for the

zeroth order ion distribution function in two different cases.

The first is that of a weakly ionized gas, the starting point

is then equation (4-20). A function $ is defined by:

f.3/2 v-w

(0) , 2

i.as i i.as ,„,,T(u; 2kTtU;Y1 I i ~ -a.asl 1 , x c5 exp{~ ~TI(ö) 1 " *i(ï)fi

a,as a,as (B-12)

With the results from the moment equations it can be shown that

-g7±M = 0. (B-13)

Equation (4-20) is now multiplied by <j>.(v) and integrated:

3*i (0)f%Y~ fiM d 3 v = 2/d3vd3£d3g6(je2 + 2g«l)I(g, SL)ta as(v+g)f 1M(v)x

m I

*• i - m i - ' i - 'o

where m = m + tn • Application of the following transforma-0 3. X

m Itions in the given order: I * -I, v + v — , g + g + I is

oequivalent to interchanging direct and inverse collisions. Withthis transformation equation (B-14) can be written as follows:

3Hi 3 3 3 (0)3Tl v g g _ g, 1 M v a > a s _ g x

m I 2

i — i *~ IDo

where: H = /f <f>2d3v. (B-16)

The conclusion is then that <t> =1 when Tj+ «, for H is non-

negative and decreases with time, i.e.:

m |v-w<0)|2

fiA} = 4 A ) ^ T I 7 Ö T ) exp(" i," (!) }- (B"17)aA

-136-

The second case to be considered is the relaxation of the

zeroth order ion-distribution function in a strongly ionized

gas. This time two equations are needed: equations (5-13) and

(5-14) of chapter V. When T Q + » they read:

a.as = (0) (1) + (1) (0) , + j (f(0) (0) ,3Tj aa a,as a,as aa a,as a,as ai a,as i,as

(B-1S)

8f ( 0 )

dti,as _ (0) r(0) . . . ,.(0) -(0) . f Q.3TJ ii i,as i,as ia I,as a,as

Again f' ' is a Maxwellian. If equation (B-18) is multiplieda y as

by (l+ln(f )) and integrated over the entire velocity spacea y as

the following equation is obtained:

->>™ - /(l+ln(f<°> ))J (f<°> fi0) )d3v1 • > *•>

where: H ( 0 ) = If. ln(f(0) )d3v. (B-21)a,as a,as a,as

The terms containing J vanish because (l+ln(f )) consistsaa a yas

of mere collision invariants. Equation (B-19) is multiplied by(l+ln(f. )) and integrated too:

x f as

»M. • "\"

Then the following inequality can be proved1:

H4,l<V«°- (B-24>(0)

from which the conclusion can be drawn that when T,+ «, f,1 ' i,as

relaxes to a Maxwellian with a temperature and a hydrodynaraic

velocity equal to those of the atoms as in the case of a WIG.

-137-

APPENDIX C

Harmonic tensors

The harmonic tensors that are used throughout this thesis are

completely equivalent to the familiar spherical harmonics, as

has been demonstrated by Johnston2. The harmonic tensor of rank

n is defined as follows:

n 2n+l

^ ^ := (2n-l)! ! VvM" (C-l)

It is an irreducible tensor, i.e. it is symmetric and a

contraction over any two indices makes the tensor equal to

zero, because v~1 is a solution of the equation of Laplace. The

harmonic tensor <y > can be seen as the irreducible part of rhe

tensor v := vvv...v (n vectors). The first few harmonic

tensors written in index notation are:

<v°> = 1, <vl>£ = v±, <v2>i:j = v l V j - f 6±y

<V3>_ _ = v v v - — (v ó.,+ v.6 + v 6 . ) • (C-2)

Any tensor can be made irreducible in a unique way, see Grad3.

One can also prove a kind of orthogonality relation, see

Wilhelm and Winkler4, which reads as follows:

2n,< h(v)>, (C-3)

where h is an arbitrary tensor of rank n and < h> is then- J n-

irreducible part of h. The following expansion is very

n-2 I n(n-l)(n-2)(n-3) i+r-r? n-41

1 + 8(2n-l)(2n-3) v të2Y ] " • • •(C-A)

where the square brackets denote the symmetric part, obtainable

by adding all the permutations and deviding the result by n!.

The inner product of v and <v > will again be an irreducible

tensor but now of rank n-1, and will thus be proportional to

<v >. This tensor also has an expansion as in (C-4).

-138-

The inner product v«<v > is thus equal to some factor times

<v >, which will appear in the right-hand side of (C-4) after

having performed the inner product with v. The tensor v«[ly ]

possesses 2(n-l)! permutations equal to v , therefore:

n 2 n-1 n(n-l) 2(n-l)! 2 n-1 ^ nv2 n-1

v.<v> = V2v - 5 ^ — — V2v + ...=—< v >

(C-5)

This result can easily be generalized to:

k n n(n-l) (n-k+1) 2k n-k- &• (2n-l)(2n-3) (2n-2k+l) v <v- >• n>k' (C"6)

With the aid of the definition (C-l) it can be shown that:

n 2n+l, n n+1 , _vVv<v > = (v<v > - <v >), (C-7)

v2

From which immediately follows with (C-5):(C-8)

If again h(v) is an irreducible tensor the following relation

holds:

v ^ V h = vn+1« h. (C-9)- - n n - - n n -With the expansion (C-4) for <v > this equation becomes:

n , n+1 , n(n+l)v2 r_ n-1 i , , ,.,v<v >• h = <v >• h + •,; ' n llv • h. (C-10)- - n n- - n n- 2(2n+l) l=- Jn n-

The fact that h is irreducible reduces this result to:n-

v < vn > . h = < v

n + 1>. h + T 2 ^ < vn ~ 1 > . 1 h. (C-ll)

- - n n- - n n- 2n+l - n-1 n- '

This relation has been employed to derive the following useful

result:

S ) = <ï >n v ̂ + ^ >n-l "2ÏT 17^v

V

-139-

The relation between the harmonic tensors and Legendre

polynomials can be Inferred from the following formula:

. , n n+1 _n .

P (Cose) = lzil_ï !_ (I), (C-13)n n! 3 n V

J 'z

where v 2 = v 2 + v 2 + v2 and v = vcos8.x y z z

Comparing (C-l) and (C-14) one obtains:

n

Let u be a vector in the direction of the z-axls; the following

result is then readily obtained:

gfr Pn(cos6) .Finally a projection operator P is defined by:

n W / V ^4irv n! S2

whih then permits the following notation for the orthogonality

property (C-3):

-140-

APPENDIX D

The Landau collision integral for identical particles

D-l The Landau collision integral

The formulation of a kinetic equation for a plasma is more

difficult than for an ordinary gas, because of the long range

of the Coulomb potential. When the number of particles in a

Debye sphere is large enough, one may use a cut-off potential.

See also chapter II. The Landau-" collision integral reads:

J „(f ,fj = C J .fG-f- Ï - - ? . ] f (v)f,(v')dV, (D-l)a8 a S aS v J= ^m v m v' ' a 8

a 3

q2q2lnA g2l - ggwhere: C „ = -2-2 , c = , g = v ' - v. (D-2)

af5 Sire2» g 3 " "o a

The following properties of G will be often used:

G«v = G»v' , G = V V g. (D-3)

If the distribution functions in (D-l) are isotropic the Landau

collision integral reduces to:

C °° 3f m 3f

a6 m v ^ ̂ v v v —^v 8 3v m v' a 3v' 'a o 3

The integration over ft , is done in spherical co-ordinates with

v along the polar axis (vv'= vv'cos6). Then:

2-rr ir j_

v~ ~r v ^ ~ivv cua a "

O O

3

, = ƒ / {v 2 + v ' 2 -2vvlcos8}2sinGd9d<)) =

Two different cases have to be distinguished:

if V < v :

if V>v : |i(^i^)

- 1 4 1 -

Straightforward differenti.itton of these expressions yields:

V 7 /gdtf . = 4IT{- iv2> + V}, if v'<v,v v ö v' ' c - ='

vD

= §J,I , if v'>v. (0-7)

Insertion of these results into (D-4) then leads to:

2C ra 9f

\z = "mf 2 I JThe functionals I and J are defined as:

P P

I ( f) _ * I ƒ xpf2f(x)dx, J = ̂ ƒ xpf2f(x)dx. (D-9)

P v p 0 p v p v

In the case of identical particles the expression (D-8) can be

written in an elegant way. If f3=oc (D-8) becomes:

2C . . 31n(f )

Two of the integrals are evaluated as follows:

In(f ) = |4itx2f dx - /4wx2f dx = n - 4ir/f x2dx, (D-lla)0 v v

00 00 3n kT »

I 2( f ) = ±2L fx»f dx - li Jx4f dx = _ 2 _ « _ *i jx«»f dx. (a v2 0 a v 2 v a m v 2 v 2 v a

aThe expression in (D-10) between braces is then equal to:

kT 91n(f ) « 31n(f )( 2 a ) f { 2 I « 3 3 }) 4Trf{x2 + I _«_(v3-X3)}f (x)dx.

^ V 3V v 3V 9V a (The second terra in (D-12) can be written in a more symmetric

form by the observation that:

3 « 3 3f » 81n(f )

-/x2fa(x)dx = f fa(v) + ƒ |-_«dx = i /(x3-v3)fa(x)^r2^x,

(D-13)

so that the collision integral when operating on isotropic

functions can be written as follows:

-142-

2C kT 31n(f )

= ——2 "5~[f (v) ln (1+ —~ 5—~) +

a a

, °° , 31n(f ) , 31n(ftiff r j 1 * 1 ré ( i-

+ — J(xJ - \

One can now see that this collision integral is zero if f is a

Maxwellian, i.e. if:

31n(f ) m m 3/2 m v 2

a ^ 2 ^ Ha a a

The collision Integral in (D-14) is still nonlinear. In the

remainder of this appendix the linearized Landau collision

integral will be investigated.

-143-

D-2 The linearized Landau collision operator for like particles

The linearized Landau collision integral for collisions

between identical particles is defined as follows:

Ja« ( f ) = Jaa ( f' faM ) + Jaa ( faM' f )' ( D" 1 6 )

where f w is the Maxwell distribution, see (D-15). If f isan

isotropic too, expression (D-10) may be used. The linearized

collision operator then reads:2C kT

|_[i (f|[i (f ) ( 1 +2 9v ° aM m v 3v

m ? aa

This operator appears in the equation for the isotropic

correction in chapter IV section 4. A generalization of (D-17)

obviously reads:

J <aa n- 'n -

C) - - 2 » ?

m va

v v'

xff(v) f(v')«<vtn> - fM(vf) f(v)«<vn>}d3v'. (D-

1 M n— n — M n— n — '

With the properties of the harmonic tensors, see appendix C,

the first part of the integrand is found to be:

Concerning the angle integration in (D-18) the following

integrals have to be calculated:

v v v v . (D-20)

The integral in the right-hand side will still be a harmonic

-144-

tensor, built up by the vector y, so that the following Ansatz

is made:

/g<v'n>dfiv, = H (v,v')<vn>. (D-21)

If on both sides of this equation the n-folded inner product

with <y > is formed the following expression for H is

obtained:

H (v,v') = (— )n/gP (cos9)dii ,, (D-22)

where 8 is the angle between v and v'. The result for n=0 was

already obtained in (D-6). After elementary integrations the

result for n=l reads as follows:

Hj = — /gcos9 dU , = -TT —(v'2- 5v 2), if v'< v,V3

= *1 C2-^'2) , if v>> v. (D-23)

For the evaluation of in the case n=l the following expressions

are useful:

Si = ( V (Hx(v,v')vj.v = ii[v'2(5 -2lll)v +^-l], if V< v,

v v U . v 2 l. if

2o - I7.I . lf vl> v« <

where QQ has already been derived in (D-7). With these

definitions the linearized operator in (D-18) for n=l reads:

C » in 3f(v')=-fVv-/f-2l'{^f(v') +i-^p—jf^Cv) +

a o a

3f(v)

i- f (v)f(v')}]v'2dv' (D-26)

oM

-145-

After some tedious but straightforward calculations this

expression is brought into the following form:

C ? m 2 2m

a a

j m 2 •

+ ' " M C T ^ ~ 3vkT J * o t f J ~ l v " ' m v 'aM 3va a a

kT kT „ ,_i_ I T / c \ I ( Ct ! J.

m v^ m v°a a

where the symbol ,J is introduced, see also equation (4-65)1 aa

Next a change of variables is made:

m v 2 kT

a a

so that for example:

111 3 /2

2kT 32f

m va

I (f) = 4irv'2(—2.) xT*"* /xC f r t 'L ; / Z f (x)dx. (D-29)

The operator ,J can then be written as follows:1 aa

f 3/2"W^ (D"30)n C kT

w h e r e : v = -2L«2_ , v | - - ^ , (D-31)2mavTa a

-146-

and the linear integro-differential operator is equal to:

2w3/2f(w) +|-(2wF(w)||]. (D-32)

One can show that JC becomes a pure differential operator by

differentiating once. After some manipulations it can be cast

into the following form:

-w 3 <„,i 32 r_ „, . -2w 32g l 3 r._, . -2w 3e ^f 3 " [2wF(w)e f J [4F(w)e ^= el 3J - 3w-

where: f = |^{e"Wg}. (D-33)

The function F(w) appearing in these expressions was defined in

(D-29) already. An important property is the following:

w , _F(w)e~W = hjx e Xdx. (D-34)

0

F satisfies the following differential problem:

| £ - F = W^, F(0) = 0, (D-35)

from which the following power series valid near w=0 is

obtained:

/ % V k+3/2 ak 1 ,„ _ , .F(w) = 2, a w , a =• , a 0 = -r . (D-36)

k=0 k K + i k + j J

An as3onptotic expansion for w + °° reads:

^/ N i/ /• x L % i v (i)r(k+^) ,n „ s

F(x) ~ j/iexp(x) - W I '|i ̂ y. (D-37)4/iT k=0 x

The special interest in the function F stems from its frequent

occurrence In the operators derived from the Landau collision

Integral.

-147-

D-3 Matrix elements for the operators obtained from the Landau

collision integral

In the chapters V and VI the Galerkin method is used to

calculate the non-isotropic parts of the electron distribution

function in a strongly ionized gas. Several integrals that are

used in this method will be evaluated in this appendix. First

the operator is treated, see (D-32). The matrix element (p,q)

for this operator is defined by:

00

X = fwPe"W/(wqe"W)dw. (D-38)pq J

Q

Straightforward substitution of X into this expression leads to

the following integrals to be calculated:

CO

T = rwme~WY(n,w)dw, m > - l , (D-39)on

o

where Y(m,n) denotes the incomplete gamma function:

w 1

Y(m,n) = Je~Xx dx, n > 0. (D-40)

The matrix elements X turn out to be:pq

ffff5/2r*- f(p+q+3/2)(p+q+5/2) }.

The function I(k) is defined by:CO

I(k) := JwkF(w)e"2wdw = h\ 3 / 2 > k > -5 /2 . (D-42)

It is of special interest because it appears also in the matrix

elements of the differential operators. The following

recurrence relations facilitate the computation of the

coefficients T :mn

-148-

T = mT . + r ( " * n ) T , + T . = r ( m ) r ( n ) . (D-43)mn m - l , n „m+n * m - l , n n- l , ra v ' v '

From these the following expressions are directly obtained:

T 0 , n = F ( n > 2 n • T mnrH = * r 2 ( - « ) , T ^ = 2 n 1 ( n + 2 ) r ( n ) ,

T = (1 -2~n~V(n+l), T, = ^ r ( n ) - 2I(n-l),ni $n

(D-44)%,n

and so on. For I(k) one easily obtains:

= kl(k-l) + r ( ^ 5 ^ } • k >

K O ) = i(|) % , I(%) = Y^- , I(-l) = iir^{ln(l+2%) - 2"^}. (D-45)

The matrix elements for the operator £ are then simply (see

also (5-79)):

\ = /wPe"Wi'(wqe~W)dw = X - /v(w)wpfqe"2wdw =pq 0 pq J

o

' = Xpq " **V(lrf-q) " Jp /wlH'q+2q(w)dW (D-46)

(see (4-106) where Q and (3 were introduced).

The matrix elements for the operator Jb are defined by:

00

6 = JwP.Z>(wq)dw. (D-47)P q o

With the definition (D-33) these are calculated as:

6pq - 2pq{((p+l)(q+l)-4)l(p+q-3) + H ^ " ^ ] , (D-48)

which clearly is a simpler expression than (D-41). The

analogous expression with the operator 3) reads:

00

pq ' i pq

-149-

+ JL- J(w-p)(w-q)wfH"qQ(w)e~Wdw, p,q > 0 . (D-49)

O

In equations (5-83) and (5-88) of chapter V it was already

demonstrated that these coefficients are symmetric for integer

values of p and q with one exception:

\ \ °°3 0 1 = -Si- + % = i-(? + I /(w-l)w

2e"WQ(w)dwJ. (D-50)4 4 P o

Finally the matrix elements for the Spitzer operator A are

given by:

CO

S r P4S Q

pq ^

{2pq+4(p+q)}l(p+q-l) + ^ ^ L + -^rdri-qfl). (!

This section is closed with the remark that in the expressions

for the matrix elements given above p and q need not be

restricted to integer values, but can take arbitrary values as

long as the integrals converge. This is an extension of the

method described in the litterature . The recurrence relations

for T and I(k) permit easy calculations of all coefficients.mn

-150-

APPENDIX E

Renormallzation of the ion multiple collision term

In section 2 of chapter IV the corrections to the electron

distribution function were obtained up to second order of e.

The ion multiple collision term in the second order

contribution is (cf. equation (4-64a)):

/o\ 2C .n} T,,x(c) .,.(2) _ _ ei lras (1)^ (1)

-e.as c 3 -e.ase

It is the solution of the following equation:

J(0)(f(2) ) = - J(?>(f(1) ), (E-2)ea e,as ei e,as

if an arbitrary isotropic function satisfying the homogeneous

equation is momentarily not taken into account. In higher order

the following equations will appear:

AV O - - 40)<C» • -»2-Only solutions proportional to c are relevant, cf. (E-l), so

that the solution in order n reads:

f ( n ) = _ 2 n i ? a s C e i T ( f ( c ) > {

e,as m c3 -e.ase

All of these contributions have a singularity at c=0, namely:

which relation is valid if f is a Maxwellian and T,,.(c)

goes to some constant value if c + 0. Thus infinitely large

contributions to the transport coefficients are obtained as a

result of a nonuniformity of the expansion in powers of e. This

divergence is removed by summation:

-151-

f(2)norm = y f(n) , _ 2Ceini,asT(l)(c) f(l)

-e.as ^ -e.as 3 + (0) -e.as'e ei i,as (l)v

( g_ 6 )

This expression gives convergent contributions to the transport

coefficients and will be used instead of (E-l).

References to the appendices

1. S.Chapman and T.G.Cowling:"The mathematical theory of non-

uniform gases", Cambridge University Press, 1970.

2. T.W.Johnston, J.Math.Phys. 7^1966)1453.

3. H.Grad, Phys.Fluids 4(1961)696.

4. J.Wilhelm and R.Winkler, Beitr.Plasmaphysik 8(1968)167.

5. L.D.Landau, Phys.Zeits.der Sowjetunion 10(1936)154.

6. M.Mitchner and C.H.Kruger:"Partially ionized gases",

J.Wiley, 1973.

Samenvatting

Het werk dat in dit proefschrift wordt gepresenteerd toont aan

dat een perturbatie ontwikkeling in het kader van het meertijd-

schalen formalisme zeer geschikt is om het gecompliceerde

stelsel vergelijkingen aan te pakken welke de transport

verschijnselen in een gedeeltelijk geïoniseerd gas

beschrijven.

De vergelijkingen beschrijven alleen elastische botsingsproces-

sen. Het doel van het huidige werk is de beschrijving van

transportverschijnselen bij energieën van thermisch niveau en

de berekening van transportcoëfficiënten. Aangezien gedeel-

telijk geïoniseerde gassen in het algemeen plasma's van lage

temperatuur zijn met thermische energieën veel lager dan het

eerste excitatieniveau is de bovengenoemde beperking niet

onrealistisch.

De basisvergelijkingen worden in hoofdstuk II gegeven. Voor de

beschrijving van de Coulorab-botsingen wordt de Landau botsings-

integraal toegepast, terwijl de andere botsingen door de

Boltzmann botsingsintegraal worden beschreven.

Diverse parameters die voorkomen, zoals het electrisch veld en

het Knudsen-getal, worden gerelateerd aan de belangrijkste

kleine parameter in het probleem: de wortel uit de electron-

atoom raassaverhouding. Ook de ionisatiegraad wordt aldus

ingeschaald en dit geeft aanleiding tot de indeling van het

gedeeltelijk geïoniseerde gas in vier gebieden van zeer zwak

tot sterk geïoniseerd.

In hoofdstuk III wordt het zeer zwak geïoniseerde gas

behandeld. Wanneer de Coulomb botsingen volledig verwaarloosd

worden beschrijft de vergelijking voor de nulde orde electronen

verdelingsfunctie de relaxatie naar een Davydov-verdeling. Dit

proces vindt plaats op de tijdschaal voor energierelaxatie

tussen electronen en atomen. De noodzaak van een isotrope

korrektie op deze verdeling wordt ook aangetoond.

Wanneer Coulomb botsingen worden meegenomen in het zeer zwak

-153-

geloniseerde gas beschrijft bovengenoemde vergelijking de

competitie tussen de tendensen naar een Davydov- en een Maxwell

verdeling. Deze vergelijking is nu niet lineair ten gevolge van

de electron-electron botsingsintegraal.

Het zwak geïoniseerde gas wordt in hoofdstuk IV behandeld. Ook

hier is een isotrope korrektie op de nulde orde electronen

verdelingsfuctie noodzakelijk. De integro-differentiaal-

vergelijking voor deze functie wordt analytisch opgelost. Het

blijkt dat er voor een gegeven electron-atoora botsingsdoorsnede

in feite zes verschillende isotrope korrektie functies zijn.

Ook verschijnen er nieuwe transport verschijnselen welke op

niet lineaire wijze afhangen van de gradiënten en krachten. De

symmetrierelaties van Onsager zijn hiervoor niet meer geldig.

In hoofstuk V wordt het sterk geïoniseerde gebied behandeld. De

integro-differentiaal-vergelijking voor het niet-isotrope deel

van de electronen verdelinges functie is in de vorm van een

vierde orde gewone DV geschreven. In de limiet van een volledig

geïoniseerd gas gaat deze zelfs over in een tweede orde DV. Dit

betekent een nuttige aanvulling op de theorie van Spitzer en

geeft eenvoudigere berekeningen voor de transportcoëfficiënten.

Numerieke berekeningen in realistische gevallen zijn samengevat

in hoofdstuk VI. De toepasbaarheid van de resultaten in het

zwak geïoniseerde gebied hangen sterk af van de gebruikte

electron-atoom botsingsdoorsnede. Voor argon blijkt het

Rarasauer minimum zware beperkingen aan de toepasbaarheid van de

theorie in te houden. Berekeningen van de transportcoëfficiën-

ten worden ook vergeleken met zogenaamde mengregels. De meng-

regel voorgesteld door Frost blijkt, vooral gezien de onnauw-

keurigheid waarmee de botsingsdoorsnedes bekend zijn, redelijk

betrouwbaar voor de berekening van transportcoëfficiënten.

Er wordt ook aandacht geschonken aan zogenaamde seeded

plasma's. Voor een cesium-seeded argon plasma is het electrisch

geleidingsverraogen berekend. Daarbij worden twee sterk van

elkaar verschillende reeksen metingen van de electron-cesium

botsingsdoorsnede vergeleken.

-154-

Nawoord

Voor de prettige samenwerking en het kritisch volgen van mijn

verrichtingen dank ik Piet Schram.

Voor de nuttige opmerkingen tijdens de besprekingen van het

manuscript wil ik ook Ties Weenink danken.

De (ex-) leden van de werkeenheid gasdynamica wil ik bedanken

voor de werkbesprekingen o.l.v. Rini van Dongen waaraan ik

mocht deelnemen.

Verder bedank ik alle leden van de vakgroep transportfysica

voor de prettige tijd die ik met hen heb beleefd in W&S.

Korte levensloop

Geboren te Eindhoven op 2 november 1952.

Middelbare schoolopleiding Atheneum-B gevolgd aan het

st.Bernardinus college te Heerlen van 1965 tot 1971.

Studie electrotechniek aan de Technische Hogeschool Eindhoven

van 1971 tot 1978.

Van 1978 tot 1982 wetenschappelijk assistent in de werkeenheid

kinetische theorie van de vakgroep transportfysica van de

afdeling natuurkunde aan de TH Eindhoven.

Stellingen behorende bij het proefschrift van

F.J.F, van Odenhoven

Eindhoven, 18 februari 1983.

Bij de zogenaamde hydraulische sprong stroomt het water van de lage naar de

hoge zijde. Dit volgt uit de energiebalans en het feit dat de entropie moet

toenemen. Deze conclusie wordt ook door Landau en Lifshitz bereikt, maar op

grond van een foutieve berekening. Zij laten namelijk de bijdrage tot de

energieflux van de potentiële energie in het gravitatieveld ten onrechte weg.

1) Landau and Lifshitz: A course of theoretical physics, vol.VI,

Fluid Mechanics, Pergamon Press, 1966, p.398.

II

De eerste orde correctie van het over een gyratieperiode gemiddelde magne-

tische moment van een geladen deeltje in een inhomogeen magnetisch veld is

in het kader van de adiabatische theorie gelijk aan nul. Dit resultaat

volgt niet uit de berekeningen van Northrop .

1) T.G.Northrop: "The adiabatic motion of charged particles",

Interscience Publishers, 1963.

Ill

In een zeer zwak geïoniseerd gas is het noodzakelijk een isotrope correctie

op de nulde orde verdelingsfunctie van de electronen toe te laten. Een bewe-

ring van Bernstein van tegengestelde strekking is derhalve onjuist.

1) I.B.Bernstein in: "Advances in Plasma Physics", vol.3, 1969, p.127.

2) Dit proefschrift, hoofdstuk III.

IV

In een instabiele schuiflaag voldoet de gradiënt-lengte van het snelheids-

profiel beter als karakteristieke lengte dan de impulsverliesdikte.

In de behandeling door de Groot et al. , van een electron-foton gas is ten

onrechte de dynamische afscherming geheel buiten beschouwing gelaten.

1) S.R. de Groot et al.: "Relativistic kinetic theory", North Holland

Pub1i shing Company,1980.

VI

De uitdrukking van Rostoker voor het tensoriële geleidingsvermogen van een

plasma is onjuist. Dit blijkt uit het feit dat 2ijn uitdrukking niet isotroop

wordt in de limiet: k -> 0. In een correcte behandeling moet met het inwendige

magnetische veld rekening gehouden worden.

I) N.Rostoker, Nuclear Fusion J_(1961)101.

Vil

De nulde orde verdelingsfunctie van de lichte deeltjessoort in een Lorentz-

gas relaxeert naar een willekeurige isotrope functie. De veronderstelling

van Chapman en Cowling dat. dit een Maxwellverdeling is, volgt niet uit de

Chapman-Enskog theorie

1) Chapman and Cowling: "The mathematical theory of non-uniform gases",

Cambridge University Press, 1970, p.188.

VIII

De in turbulentie-theorieën vaak gemaakte veronderstelling dat het ensemble

van dynamische systemen uniform is , blijkt soms in strijd te zijn met de

dynamica van die systemen .

1) R.C.Davidson: "Methods in nonlinear plasma theory", Academic Press, 1972.

2) I.E.Alber, Proc.R.Soc.Lond. A363(1978)525.

IX

Oplossingen van de electronentemperatuurvergelijking duiden er op dat

macroscopische "runaway" van electronen in een gedeeltelijk geïoniseerd

gas slechts in uitzonderlijke omstandigheden te verwachten valt.

1) Dit proefschrift, hoofdstuk V.

I'te Met betrekking tot de evenwichtige opbouw van onderzoek- en onderwijs-

>'- programma's is het wenselijk om bij de afsluiting van onderzoekcontracten

• ;. met bedrijven een «xtra percentage in rekening te brengen voor gelieerd'•'-', onderzoek van fundamentele aard.