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Kinetic theory of transport processes in partially ionizedgasesCitation for published version (APA):Odenhoven, van, F. J. F. (1983). Kinetic theory of transport processes in partially ionized gases. Eindhoven:Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR82292
DOI:10.6100/IR82292
Document status and date:Published: 01/01/1983
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F.J.F. van Odenhoven
KINETIC THEORY OF TRANSPORT PROCESSES
. IN PARTIALLY IONIZED GASES
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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
Page 4
DIT PROEFSCHRIFT IS GOEOGEKEURD
DOOR DE PROMOTOREN
PROF.OR.IR. P.P.J.M. SCHRAM
EN
PROF.DR. M.P.H. WEENINK
Page 5
Wees niet bang voor het
langzaam voorwaarts gaan,
wees slechts bevreesd
voor het blijven staan
(Chinees gezegde)
Page 6
Contents
page
I Introduction l
References 5
II Basic equations 6
References 15
III Very weakly ionized gases 16
17
IV
v
VI
III-1 The electron distribution function
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
Weakly ionized gases
IV-1 Heavy particle results
IV-2 Perturbation solution of the electron
distribution function
IV-3 The macroscopic electron equations
IV-4 The first order isotropic correction
IV-5 Electron transport coefficients
IV-6 Modifications for a seeded plasma
References
Strongly ionized gases
V-1 Heavy particle results
V-2 The electron kinetic equation
V-3 The electron macroscopic equations
V-4 The nonisotropic part of the
electron distribution
References
Numerical results
VI-1 The isotropic correction
VI-2 Electron transport coefficients
37
38
39
46
51
57
61
67
69
70
72
77
82
,86
93
94
95
108
Page 7
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-1 Electron-atom collisions
A-2 Electron-ion co11is.i.ons
A-3 Moments of the electron-heavy particle
collision integral
B Some H-theorems and properties of collision intesrals
B-1 The zeroth order electron-atom
collision operator
B-2 The zeroth order electron-ion
collision integral
B-3 Ii-theorems for the ion distribution function
c Harmonic tensors
D The Landau collision intesral for identical particles
D-1 The Landau collision integral
D-2 The linearized Landau collision operator
for like particles
D-3 Matrix elements for the operators obtained from
the Landau collision integral
E Renormalization of the ion multiple collision term
References to the appendices
Samenvatting
Nawoord
Korte levensloop
124
128
130
133
134
135
137
140
143
147
150
151
.152
154
154
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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 1.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 textbooks3, 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 3•
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 devices. 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 Maxwellian 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.
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In the fully ionized or Spitzer limit the isotropic part of the
electron distribution function is a Maxwellian and the non
isotropic part has been computed numerically by Spitzer and
HMrm7• 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
functions. 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:
e = (m /m )'2 e a
(1-1)
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 Weakly Nonlinear Weakly Ionized Strongly Ionized Ionized Gas Region Gas Gas
\I 2
\I
= bcr:h \I
~ « ee ee = 0(e) > 1 € --\I \I \I vea ea ea 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 Water 10 is assumed. Some
results additional to his are obtained.
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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 \i.Qrk in the field of transport coefficients in partially
inni7.Prl ~~ses was motivated by the possibility of direct energy
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conversion by means of an MHD-generator 11 • 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(n) of the relevant kinetic
equation in each order is found in terms of an expansion into
harmonic tensors (see appendix C):
f(E) f(O)(c) +
+ e(f(l)(c) + f(l)(c)•£ ) +
+e2(£(2)(c) + f(2)(c)•£ + £(2)(c):<££>) +
+ ........ (1-2)
where c is the peculiar velocity, <££> is the harmonic tensor
of second rank and 7Cn) 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 parts in higher order. The expansion
generally used in the litterature is a two-term expansion of
the form:
(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 £ and describes both fast and
slow transport phenomena by means of the multiple time scales
formalism.
Page 12
-5-
References
1. S.Chapman,Phil.Trans.R.Soc.,216(1916)279,3.!.Z.(1917)118,
Proc.Roy.Soc.,A98(1916)1.
D.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 Gasen von mittleren
Druck",in:Handbuch der Physik, Springer 1958.
C.Cercignani:"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,~(1963)332,380.
E .A.Frieman,J .Math.Phys. ,i(l 963 )410.
J.E.McCune,T.F.Morse and G.Sandri:Rar.Gas Dynam.];_(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.H~rm, Phys.Rev.,~(1953)977.
8. B.Davydov,Phys.Zeits.der Sowjetunion,!(1935)59.
9. M.J.Druyvesteyn,Physica,.!.Q.(1930)61,];_(1934)1003.
10. w.van de Water,Physica,85C(l977)377.
11. M.Mitchner and C.H.Kruger:"Partially ionized gases",J.Wiley
1973.
Page 13
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II BASIC EQUATIONS
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 Saha'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 terms in the Boltzmann equations of the three compo
nents consist of a sum over all possible elastic collisions
that may.occur:
()f s + v•Vf + .!.... F •Vvfs = ~ Jst(fs,ft). at - s m -s l s t=e,i,a
(2-1)
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 ~s' for example external forces or as
a result of a self-consistent electric field. The right-hand
side of equation (2-1) describes the variation of fs caused by
all possible elastic collisions.
Macroscopic quantities appear as so-called moments of the
distribution function fs. Important quantities are:
the density ns, the hydrodynamic velocity in the laboratory
frame w , the temperature T , the pressure P and the thermal -s s =s
heat flux Ss· These are defined as follows:
n (r,t) s -
ff (r,v,t)d3v, n w (r,t) = fvf (r,v,t)d3v, s - - s-s - - s - -
Im c c f (r,v,t)d3v, gs(E,t) = f\m2c c f (r,v,t)d3v, s-s-s s - - s-s-s s - -
Page 14
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3 -.2 n kT (r,t) ='[\m c2f (r,v,t)d3v, s s - s s s - - (2-2)
where the peculiar velocity ~s = y - !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, msy' and \msv 2 the moment equations are
the conservation equations for the particle number density,
momentum and energy respectively:
an __ s + V•(n w ) = O, at s-s
aw m n { ~-t + (w •V)w } + V•P s s a -s -s =s - n F s-s
(2-3a)
J\m v2{ I J (f ,f )Jd3v, s t*s st s t (2-3c)
In the energy equation the following notation was introduced:
e = l kT + Lm w2. ~s 2 s ~ s s (2-4)
The conservation equatlon 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 invariantsl-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:
3 <lT .
- n k{-s + w •VT } + V•g + P : Vw = J11m c 2 I J d3v, 2 S at -s S S =s -S S S t*S St
(2-5)
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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:
w -m
i:: m n w s s s-s i:: m n s s s
(2-6)
It is possible to define diffusion velocities ~s with respect
to this plasma-velocity:
u :- w - w • -s -s -m
(2-7)
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 ~s are defined:
u :== w - w • -s -s -a
(2-8)
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
fs 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
,Boltzmann collision integral:
m t m t J (f ,ft) -st s
2fd3 td3go(t2+z,...•t){f (v- t-+m )f (v+<>+ s+m- ) + Q - s - m t - Q' m
t s t s
-fs(!)ft(y+~)}. (2-9)
Here g • v - v is equal to the relative velocity just before a - -t -
collision. The validity of the Boltzmann collision integral is
based on the smallness of the number of particles in a sphere
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with radius equal to the characteristic range of the potential,
i.e. the Boltzmann 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:
v' == v m JI. t-
m +m' s t
v' -t
m JI. v + + ....!::._ a m +m
s t (2-;-10)
where ! = s'- s denotes the difference in relative velocities
just before and after a collision. The factor I(g,JI.) is the
differential cross section and is defined as:
b I flbl I(g,JI.) = cr(g,x) = sinx ax , (2-11)
where b is the impact parameter and x is the scattering angle.
It contains the geometry of the collision. The 6-Dirac function
with argument Jl.2+2a•! assures energy conservation.
Collisions between charged particles are more difficult to
treat because of the 1/r potential. The Landau collision
1ntegral3 will be used, which can be obtained from the
Boltzmann 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:
g2I-s~ C 'I •f(-"'-)•{l 'I - l 'I }f (v)f (v )d3v • st v g3 ms v mt vt s - t -t 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 est are given by:
q 2q 2lnA s t
Cs t = _8..;;.11_e:..;..2m--
o s
(2-13)
where qs and qt are the charges of the collision partners and
Page 17
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lnA is the so-called Coulomb logarithm. Herein A is the inverse
of the plasmaparameter E , and is proportional to the number of p
electrons in a sphere with radius equal to the Debye lenght r0
:
A=..!. E
p (2-14)
In a plasma one distinguishes three characteristic lenghts: the
Debye length ~· 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 interparticle distance r and 0
the Landau lenght rL, which is the distance of closest approach
between two like charged particles with thermal velocities.
These lenghts are defined as:
r 0
n-113, r D
e kT 'a (-0-J . ne 2
(2-15)
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
of f'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 r1 and r0 as integration boundaries. This leads to the
factor lnA. This factor has to be much greater than unity.
Speaking in more physical terms one could say that the Landau
lenght is so small that there are relatively very few short
range collisions. 5ecause 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 Boltzmann equation will be considered in more
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detail. To solve this complicktted equation an expansion into a
small parameter e; will be used, e; being the square root of the
electron-atom mass ratio:
e; • (m /m ) \. e a
(2-16)
This choice seems obvious and the next step is that all
dimensionless numbers, obtainable from the dimensionless
electron Boltzmann equation, are expressed as powers of e;. The
equations will, however, not be made dimensionless. All terms
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 Boltzmann equation reads as
follows:
()f ~ + e;v•Vf - e; - •V f - wce(~x!?) •Vvfe • eJ + J + e:J • , at - e m v e ee ea ei e
(2-17) wherein b is a unit vector in the direction of a constant
external magnetic field B. The electron cyclotron frequency: eB -
wee • -;-- , has been taken of the order of the electron-atom e
collision frequency:
(l) T •(/(l), ce ea
(2-18)
Here T is the mean collision time between two successive ea collisions of an electron with a neutral atom:
l T ea
n v Q(l) vTe a Te ea = ->..
ea (2-19)
Thermal velocities are defined as v •(kT /m )~and Q(l) is the Ts s s st
elastic collision cross section for momentum transfer of
particles s with particles t defined as follows:
(2-20) 0
where g is the relative speed of the colliding particles.
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In expressions like (2-19) some characteristic value for g will
be substituted e.g. vTe" Furthermore the mean free path Aea has
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:
(J(£). (2-21)
Concerning the inhomogeneities the Knudsen number ae defined as
the ratio of Aea to some macroscopic length scale L reads:
A ea (J ae := L = (<-), (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 900 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
£:
J ee J ea
v ee v ea
(1) nevTeQee
n v Q(l) a Te ea
(J ( £).
The same holds for the electron-ion collision integral.
(2-23)
A strongly ionized gas will be defined as a plasma in which the
collision frequencies satisfy the conditions: vea ~ vee ~ v .• ei
Page 20
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Next the heavy particle Boltzmann equations have to be
considered. For a weakly ionized gas one obtains:
(2-24)
3fi eE at + e: 2y•l7fi + e: 2 (m~ + wci~x!?) •\7vfi .. e;ltJie+ e:Jia+ e;
2JU • (2-25)
Some ex·tra assumptions have been made in these equations. The
time variable has been scaled with 'ea' so that in these equa
tions the choice v ~ v ~ v /e; has been made. The heavy ea aa ia
particle electron collision terms receive an additional factor
e:2 because of the fact that momentum transfer in these colli
sions is rather inefficient. From these assumptions it follows (1) (1) (1) .
that Qea ~ e:Qaa ~ Qia , which is reasonable provided that
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 vTi ~ vTa ~ e:vTe' In the chapters to follow solutions of
kinetic equations will be found by means of a perturbation
expansion:
(2-26)
It is known that such an expansion may often lead to secular
behaviour, i.e. it contains terms f n+l and f such that the s, s,n ratio f n+l/f goes to infinity with increasing time, so
s, s, n that the expansion fails. One possibility to avoid these
secularities is to make use of the multiple time scale forma
lism4-7. For that purpose it ls observed that there are
different time scales to be distinguished: t 0 is called the
fastest time scale which is connected with the mean free time
between two successive collisions of an electron with an atom;
t 0 ~ 'ea • Then successive time scales are defined in the
following manner: t 1 = t 0/e:, t 2 = t 0/e 2 etc. The t 2 time scale
will appear to be the timescale on which energy relaxation
Page 21
}
-14-
between electrons and atoms takes place. In the multiple time
scale formalism new time variables T are defined as follows:, n
n Tn := E t, (2-27)
so that the time derivative transforms as:
(2-28)
Thus the formalism consists' of a transformation from one time
variable to a certain number of time variables T whith are n
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,-r 0,1 1 , •• ) + f (r,v,1 0,1 1 , •• ) + •.. s - - s - - s - -
(2-29)
The procedure is then as follows: the collision integrals are
also expanded in powers of E 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 si The conservatio~ 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.I
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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.Zeits.der Sowjetunion,10(1936)154.
4. G.Sandri,Ann.Phys.24(1963)332,380.
5. E.A.Frieman:J.Math.Phys._i(1963)410.
6. J.E.McCune,G.Sandri and E.A.Frieman,
in Rar.Gas Dynam.! (1963)102.
7. G.Sandri,in:"Nonlinear partial differential equations",
ed.W.F.Ames, 1967.
Page 23
-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 Waterl. In the
following two sections a similar type of analysis is given for
a different ordering of some parameters. In~omogeneities 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,
In 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 Ma:Kwell
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 e 2 • 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.
Page 24
-17-
III-1 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 Jea is thus
retained in the right-hand side of equation (2-17). The heavy
atoms possess a local Maxwellian: 2
mJy-!!a (I• t) I 2kT (r t) },
a -•
where the macroscopic quantities obey the Euler equations:
an ot a + v'-(na~a) = o,
aw m n (~-a+ (w •V)w ) + Vpa O, a a at -a -a
The Mach number is assumed to be of the order unity:
=l?(l).
(3-1)
(3-2)
(3-3)
(3-4)
(3-5)
From equation (3-3) the instationary inertial term is estimated
by means of the pressure term:
(3-6)
If the electron and atom temperatures are of the same order and
a velocity transformation is applied according to:
~ + c = v - w (r,t), - - -a -
(3-7)
the electron Boltzmann equation takes the following form:
3f eE ow ~+ ec•Vf + e: 2w •Vf - {e: _..::+ e:3( -a +(w •V)w) + s2(c•V)w at - e -a e m at -a -a - -a
e
+e:w wxb}•Vf -w c-(bxVf)=J (f), ce-a - c e ce- - c e ea e (3-8)
Page 25
-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 e
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):
of(O) e w c•(bxV f(O)) = J(O)(f(O)).
ci: 0 - ce- - c e ea e (3-9)
It is possible to derive an H-theorem from this equation. In
velocity space a spherical co-ordinate
I
I
' ' I 'I
fig. 3-1.
system with cz directed along the unit
vector b is introduced. See fig. 3-1.
Equation (3-9) then reads:
af<0 > a£< 0 > ~ + w e = Je(Oa)(fe(O» • (3-10) ai:o ce 1'$
Multiplication of this equation by
(l+ln(f(O))) and an integration over e
the entire velocity space results in:
(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 i: 0 + w, since that is the
general solution of the equation J(O)(f) = o. ea
The first order part of equation (3-8) reads:
(lf(O) (lf(l) _e +a{ + c•Vf(O)_ t~ + w w xb )•Vf(O)_ w c-(bxV f(l» 3i: 1 T0 - e me ce-a - e ce- - c e
(3-12)
In a formal, procedure one may separate the distribution
Page 26
-19-
functions in an asymptotic part on the t 0 time scale and a
remaining transient part:
/0) + f(O) e,as e,t
f(O) e,as
lim f(O). e
Then equation (3-12) is integrated with respect to t 0 :
(3-13)
ilf(O) ~ {- _e,as + ( (1) ) ' w c • bx'J f O ar 1 ce- - c e,as
eE' T ilf(O) - c•'Jf +.....:.. •'J f(O) + J(O)(f(l) >}+Jo{- ~,t - c•'Jf(O)
e,as me c e,as , ea e,as 0
ar 1 e, t
eE' + w C•(bx'Jf(l)) +.....:.. •'J f(O) + J(O)(f(l))}d~
ce- - e,t m c e,t ea e,t O• e (3-14)
where E' • ~ + ~ax~. (3-15)
If it is assumed that the integral in this equation remains
finite when r 0+ ""• the first part in the right-hand side would
increase without bounds with t0
except if it is demanded that:
ilf(O) eE' _e,as+ c•'Jf(O) _.....:.. •'J f(O) <lt 1 - e,as me c e,as
/O)(/l) ) + w c•(bx'J f(l) ) • ea e,as ce- - c e,as
(3-16)
This equation can be solved easily if f(l) is expanded in / e,as
harmonic tensors:
f(l) e,as
= f(l) (c) + f(l) (c)oc + f(l) (c) -<cc> + •••• e,as -e,as - =e,as -- (3-17)
Insertion of this expansion in equation (3-16) then gives with
the aid of appendix A and definition (4-61) for ~(n): .., l {( 1 ( ) - nw bx) f(l) (c) }-<c~ ::: l M • f(l) (c)-<c~ •
n=l t(n) c ce- n-e,as n - n •(n) n-e,as n -
ilf(O) e,as
]Tl = o.
eE'(lf(O) (
- e as =£·;-ac> -e
'Jf(O) ) e,as ' (3-18a)
(3-18b)
The latter equation is the isotropic part of equation (3-16).
Page 27
-20-
From the right-hand side of (3-18a) it appears that n=l gives
the only
fying the
contribution apart from an isotropic function f satis
homogeneous equation:
/O)(f) + ea
Thus the solution for f(l) reads: e,as
(3-19)
f(l) (c) = f(l) (c) + eE''ilf(O)
( ) ,.... l r - e , as ,., ( 0) ) e,as - e,as \1) c !:•g(l) • 'm Tc - vfe,as '
e (3-20)
In second order equation (3-8) yields:
For reasons of simplicity this equation will be dealt with in
the limit ,0
+ 00 • The isotropic part can easily be separated
from the rest by means of the otho3onality property of the
harmonic tensors (see appendix C)
+ w •'ilf(O) -a e,as
(3-22)
This equation may be integrated over T 1, if !a 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:
(3-23)
With the results in appendix A and expression (3-20) for f(l) -e,as
equation (3-22) ls written as follows:
Page 28
-21-
af<0 > af< 0> _e,as + w •Vf(O) _ .£. _e,asV•w = l_.A.'.(c3'T: M:l •Jl'if(O) ) + d'T:z -a e,as 3 ac -a 3c - (1)=(1) - e,as
m 3 kT + _e_ ~{S...... L'l + a a )f(O) }
m c2 Cle \1) mec ac e,as , a
(3-24)
' where.!'!_' = V - ~-~. This. equation has been derived earlier
m c Cle e
by Bernstein2 and 0ien3 • An isotropic correction is not
mentioned by these authors. The non-isotropic part of (3-21)
when , 0+ 00 reads:
c•Vf(l) + <cc>:Vf(l) e,as -e,as
(lf(O) -<cc>:Vw .l~e,as
-a c ac w c•(bxV f(Z) )
ce- - c e,as
eE' ilf(l) _ <cc>: - -e,as
mec ac
+ /0) (f(2) ). e,as e,as (3-25)
Insettion of an expansion like (3-17) for
following solution of equation (3-25):
f(Z) leads to the e,as
f(Z) (c) = f(Z) (c) + c•f(Z) (c) + <cc>:f(Z) (c) e,as - e,as - -e,as · -- =e,as '
f(Z) (c) -e,as
- ( )',- 1- • " '~f ( 1 ) '(1) c g(l) "'::: e,as'
df(O)
f(2) (c) = - ' (c)M:l •(clt'f(l) 1 e,asVw ) =e,as (2) =(2) - -e,as - cac -a,as,
where the isotropic part 1(2) is as yet undetermined. e,as
(3-26)
(3-26a)
(3-26b)
The third order part of the electron Boltzmann equation (3-8)
is:
d a where dt =Ti:+ !a •V.
The isotropic part of this equation can again be separated from
Page 29
-22-
the rest. When To+ m this isotropic part yields the following
equation for the first order isotropic correction:
af(O) a!(l) a1< 2) a1< 1) _e,as + _e,as + _e,as + £:.\,.f(2) + w •Vf(l) _ £ _e,asV•w OT3 OT2 dTl . 3 -e,as -a e,as 3 ac -a
eE'
3m c 2 e
(3-28)
Insertion of expression (3-26a) for f( 2) and using appendix A -e,as
then gives:
....!..JP•{c 3T ~r,l •A'f(l)} + 3c - (l)=(l) - e,as m c
a
Equation (3-29) may be integrated over T1• Elimination of
secular behaviour then leads to the following equations:
o, (3-30a)
a1< 1> a1< 1> + e,as + w •Vf(l) _ £~e,asV•w =
r.2 -a e,as 3 ac -a
m 3 kT = !... tA.' • {c3T M:l •J!.'·f(l) } + ....L _L{£._(1+ __!. L)f'(l) }•
3c - (l)=(l) - e as m c2 ac T m c ac e as ' a (1) e ' (1) (3-30b)
The latter equation for f is almost equal to equation (O) e,as
(3-24b) for f , which is homogeneous. Equation (3-30b) {las a e,as source term containing the zeroth order.distribution function.
These equations are different from the corresponding equations
of van de Waterl, due to the different ordering.
The inhomogeneity of equation (3-30b) obstructs the absorbtion
of the first order isotropic correction into the zeroth order
distribution function, which
Bernstein2. The equation for
variable c. In the following
was an assumption made by
f(l) is of second order in the e,as
section two conditions will be
given which determine the two constants of integration.
Page 30
-23-
III-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 s~bsequent integration
over the entire velocity space. The following equations are
then obtained:
an _e + e:V•(n u ) + e:2V•(n w ) = 0, at ·e-e e-a (3-31)
au dw m n { ... -e + e:(u •V)u + e;2(w •V)u } +e:2m n (u <''J)w + e;3m n ..::_a+
e e ot -e -e -a -e e e -e -a e edt
e:V•P + e:en E + m n w (u + e:w )xb = fm cJ (f ,f )d3c, =e e- e e ce -e -a - e- ea e a (3-32)
al n (-e + e:u •Vl + e;2w •VE: ) + e:Vo(9. + P •u ) + e:en u •E + e at -e e -a e e =e -e e-e -
dw + e:mene~e·(~ax~) + e: 3mene~e·d~a + (P+mnuu):Vw + =e e e-e-e -a
(3-33)
Note the transformation that has been made according to (3-7).
Therefore ie is now defined slightly different from (2-4) as:
l = 12 kT + ~m u2. (3-34) e e e e
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 orde:t of e::
(3-35)
(3-36)
In first order one obtains:
(3-37)
Page 31
-24-
+ (O)E' (0) en •u e - -e o. (3-39)
When T0+ ~ 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(O) and the heat flux g(O) vanish. _e e
Equations (3-37) and (3-39) take the following form when T0+ m:
an(O) ar(O) e,as e,as = O,
hl .. hl
en(O) (E'+ u(l) xB) + e,as - -e,as -
m c V (0) + J e-( )f(l) d3c, Pe,as '(l) c e,as
where p(O) = n(O) kT(O) • e,as e,as e,as
(3-40)
(3-41)
Thus it is seen that many terms in these equations vanish when
T0+ ~. The expression for f(l) found in (3-20) may be e,as
substituted into equation (3-41) which then yields an identity.
The second order equations are given in the limit T0+ 00 in
order to reduce the complexity of the equations:
an(O) an(l) e,as + ~e,as + V•{n(O) (u(l) +~a)}= O, "li'T2
oT1
e,as -e,as
au(l) (0) ( -e,as + (2) b) + V•u(l) + en(l) E' + m n - wceYe asx- "-e e,as aT
1 , -e,as e,as-
m c +I~ f<2> d3c
T(l)(c) e,as 0,
(3-42)
(3-43)
dT(O) OT(l) 3 (0) k( e,as + _e,as + Y(l) •VT(O) )+ V•( (1) + E(O) • (1) ) "211e,as "QT2 aT
1 e,as e,as ge,as -e,as Ye,as
Page 32
I /
-25-
+ en(O) E'•u(l) + P(O) :Vw e,as- -e,as =e,as -a
m m c2 - ....!:..J e (1+ l...)f(O) d3c.
ma T(l)(c) mec ac e,as
(3-44)
From equations (3-18a), (3-20) and (3-24a) it is inferred that:
= o. (3-45)
It will be assumed now that the following first order quanti
ties are zero:
(1) n e,as
= T(l) e,as o, (3-46)
which are the additional conditions needed for a unique solu
tion of equation (3-29). Such condi·tions 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 €· Since
moreover f(O) is isotropic the second order equations now e,as
reduce to:
an(O) ~e,as + V•{n(O} {u(l) +~a)}= O, oT 2 e,as -e,as (3-47)
m c en(O) u(Z) xB + }--=.:...._ f(Z) d3c = O,
e,as-e,as - '(l)(c) e,as (3-48).
dT(O) 3 (0) k{~e,as + u(l) •VT(O) } + V•(g(l) + P(O) u(l) ) + "T1e,as d, 2 -e,as e,as e,as e,as-e,as
+ en(O) u(l) •E' + p(O) V•w e,as-e,as - e,as -a
(3-49) (0)
As all quantities occurring here are functionals of f , see e,as equation (3-20), (3-26) and (3-29), it appears that these equa-
tions do not contain any variations with T1
, so that the t 1 time scale has no physical meaning in this particular sttua
tion. Insertion of expression (3-26) for f (Z) into (3-48) e,as
leads to:
Jc2A'f(l) a3c = o. - e_,as
This equation can be further evaluated to give:
(3-50)
Page 33
-26-
Vp(l) + en(l) E' = 0 e,as e,as- ' (3-51)
which is satisfied through the requirements (3-46).
With the aid of equation (3-47) the energy equation can be
written in the following form:
- p(O) £__ ln{n(O) (T(O) )-3/2} + u(l) •(en(O) E' e,as DT 2 e,as e,as -e,as e,as-
+ Vp(O) ) + e,as
m m c 2 kT + V•g(l) - ._!. f-e- (1+ ~~)f(O) d3c
e,as ma T(l)(c) mec ac e,as ' (3-52)
D d (1) a where: - = - + u • V = - + DT2 dT2 -e,as dT2
(w + }l) ) •V. -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(O) u(l) e,as-e,as
where
- ~ g(l).~, + V•(n~~~s~(l)),
2 3f(O) (1) e l ( )'.-1 . e,asd3
g = - 3m cT(1) c ~(1)ac c, e
1
3n(O) 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(O) e,as
= n(O) f (c) e,as o '
(3-53)
(3-54)
(3-55)
(3-56)
where f (c) depends on c only, so that the space and time
depende~cies occur through n(O) solely. With this assumption a , e,as diffusion equation may be obtained from equation (3-47):
<ln(O) _e,as + n<I>.vv (O) _ .!. (l)·E''Vl ( (0) ) = 0 <lT 2 ~ • ne,as e g ·- n ne,as ' (3-57)
where the neutral component has been assumed to be homogeneous
Page 34
-27-
in space. The assumption in (3-56) also implies a uniform
electron temperature. Refinements can be obtained by making an
expansion of f(O) in the spatial derivatives of n(O) • see e,as e,as'
e.g. reference 4.
These equations are used for the determination of electron-atom
cross sections from diffusion experimentsS.
The thermal heat flux is also calculated with the aid of
expression (3-20):
n(l) g(l)•E'- ~(l)•Vln(T(O) ) ~e,as -q - - e,as
+ V•(n D(l)) e,as=q '
e kT(O) m c2 af (O)
(3-58)
with: __ e_.,..;..a..;..s_ f( e . SJ -1 e,asd3 3m 2kT(O) - 2 CT(l)~(l)ac c,
e e,as (3-59)
(3-60)
(3-61)
It appears from expression (3-26a) for f( 2) that corrections -e,as
to the transport coefficients are given by the same expressions
if f(O) is replaced by 7(l) • 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
second order diffusion velocity u( 2) vanishes. The second -e,as order thermal heat flux reduces in this special case to:
(3-62)
The first order fluxes reduce to the following expressions in
case of Maxwell interaction between electrons and atoms:
u(l) -e,as
eT kT(O) - .:.:ill. M: 1 • (E' + ~Vln( ( 0) ) J
me =(l) - e Pe,as ' (3-63a)
(1) 9.e,as
ST n(O) (kT(O) ) 2 (1) e,as e,as M:l •Vln(T(O) )
2me =(l) . e,as '
Page 35
-28-
showing that there are no cross effects in this case.
In third order of e the moment equations, when considered
asymptotically on the t 0-time scale, read:
(3-64)
au(l) au<2> dw m n(O) {....:.e,as + ....:.e,as +(u(l) + w )•Vu(l) + ( (1) V) + -a}
e e,as at2 ot 1 -e,as -a -e,as ~e,as· ~a dt
(2) (2) (2) (1) + V•P + en E + m n w (u + w )xb = =e,as e,as- e e,as ce -e,as -a -
(3-65)
at<0 > aE< 2 > n(O) (-e,as + . e,as + u(2) •V[/O) ) + V ( (2) + E(O) (2) ) e,as at 3 lt1 -e,as e,as • ge,as -e,as·Ye,as
(0) (2) (0) (2) + en u •E' + m n w u •(w xb) e,as-e,as - e e,as ce-e,as -a -
m 3kT m c2 . • ...!.J(-a - _e_ + kT c B-1-) )f(l) d3c.
ma r(l) r(l) a ac t(l) e,as (3-66)
Again an Ansatz is made, namely:
n(2) e,as
= T(2) e,as o, (3-67)
which can be justified in the same manner as in (3-46).
Equations (3-64) and (3-66) may then be written as follows:
(3-68)
- p(O) (.!_+ u( 2) •V)ln{n(O) (T(O) )-3/2} + en(O) u( 2) •E' + e,as at3 -e,as e,as e,as e,as-e,as -
17 (0) Pe,as (3-69)
From (3-23) and (3-26a) it can be deduced that:
Page 36
3u(Z) _:.e,as = O. 3'£1
-29-
Equation (3-65) may therefore be written as follows:
Du(l) m n(O) (_:.e,as
e e,as DT 2
Dw +_:.a) + V•P(2)
Dt =e,as
m2c2 '·T ~ "'a'(l) a f e {1- _ __,_ _ _,__ -(c'+
3ma't:(l) 2m c4 ac e
where~=~+ u(l) •V. Dt dt -e,as
(3-70)
(3-71)
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.
Page 37
-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:
* in which: Ta = Ta +
2 ma(eEi:(l)(c))
3km e
(3-72)
(3-73)
is a function of c. The relevant macroscopic equations read:
an(O) e,as = O,
a:t2 3"'(0)
3 (0) k '"e,as ~e.as ai:2
Equation (3-72) may be solved by means of the method of
s~paration of variables. Insertion of
f(O) e,as
into equation (3-72) results in the following eigenvalue
problem for the function f:
* m kT _e_ .!!_{c3-1-(f + ___!!. i!..J} + Af(c) = O, m c2 de '(l) mec de
a
and a simple equation for the function h:
dh + :>.h = o. di:2
If 1=0, equation (3-77) can be directly integrated. The
solution y0 , the eigenfunction for :>.=O, then reads:
c m c'dc' Yo= A exp{- f e* }.
o kTa(c')
(3-74)
(3-75)
(3-76)
(3-77)
(3-78)
(3-79)
Page 38
-31-
This is the asymptotic solution of (3-72) when 12+ 00 , and is
known as the Davydov distribution function 6• It will be demon
strated now that all other eigenvalues are positive.
Define:
f(c) = y0(c)$(c)~ (3-80)
Substitution into (3-77) and subsequent multiplication by $ and
integration then leads to: "" 2 J p(c)y0(c)(2fdd ) de 0 c A = ...;;..~~~~~~~~
00
J y0(c)$2(c)c 2dc 0
where y 0(c) and p(c) =
(3-81)
(3-82)
are positive functions, so that all eigenvalues except,A=O are
positive indeed. Expression (3-81) also gives a device for the
calculation of the eigenvalues and eigenfunctions by means of a
variational principle. From equation (3-77) one can deduce that
all eigenvalues are orthogonal with weighting function c 2y 0 :
f y 0c 2$ $ de = D, n*1n. nm 0
The variational principle then reads as follows:
(3-83)
.A =min R($) = R($ ); fy 0(c)$ (c)Q> (c)c 2dc = O, m=O,l, ••• ,n-1. n n
0 n m
where: R($)
fy 0(c)$2(c)c 2dc 0
(3-84)
(3-84a)
A Rayleigh-Ritz method may be used to approximate the first N
eigenvalues and eigenfunctions. In the special case of Maxwell
interaction between electrons and atoms the eigenvalue equation
can be solved ~irectly. Then the collision time '(l) is a
constant, so Ta does not depend on c either. The eigenvalue
equation after a transformation of variables reads:
Page 39
-32-
d2cb 3 ~ >. ~+ <z - w)dw + 2'(1)~(w) 0, (3-85)
m w2 e where w = --* . Equation (3-.85) is the differential equation
2kT a of Laguerre. The eigenvalues and eigenfunctions are thus equal
to:
2n '-n = '(1)'
n=O, 1, 2, ••••• (3-86)
The Davydov distribution function is now a Maxwellian with
* temperature equal to Ta.
In the case of a hard spheres interaction model one has:
!l =-c· (3-87)
where !l is a constant mean free path. A straightforward calcu-
lation shows that the Davydov distribution is now equal to:
C ( 2)(1 + ~)aA me A y 0 = exp -ac A , a = 2kT , = a
where the constant C is fixed by:
m (!leE) 2 a (3-88)
(3-89)
In the cold gas limit Ta+ O, the Druyvesteyn distribution7 is
recovered:
Yo = C exp(-yc4 ), y 3m3
e
4111 (R.eE)2 a
(3-90)
If the eigenvalues and eigenfunctions are known, the initial
value problem may be solved, i.e. equation (3-72) supplemented
by the condition:
f(O) (c 0) = n(O) f (c). e,as ' e,as 0
(3-91)
The formal solution reads:
"" f(O) (c T ) = E n(O) a y 0(c)~ (c)exp(->. <2), e,as ' 2 n=O e,as n n n
(3-92)
Page 40
with: a n
ff 0(c)$n(c)c2dc, 0
-33-
if the eigenfunctions are orthonormal: .. f4> 2(c)y0(c)c 2dc = 1, n=0,1,2, ••• o n
and form a complete set.
(3-93)
(3-94)
The same problem has been investigated by Braglia et alB, who
calculated the temporal behaviour of the distribution function
for various cross sections.
Page 41
,-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 m /m , i.e.: e a n Q(l)
e ee n Q(l) =
a ea
(3-95)
the e-e and e-i collision terms appear in the second order
equation of section 1. When T0+ ~. only the isotropic part
changes, and the equation for the zeroth order electron distri
bution function now becomes a nonlinear integro-dif f erential
equation:
af(O) _e,as + w •Vf (O) 3T -a e,as
af(O) - _S:. _e,asV•w = .!_JI.' •(c3T M:l •..t' f(O) )
3 ac -a 3c - (l)=(l) - e,as 2
+ meJ:-(_£_ (l+ kTa L)f(O) ) + J (f(O) f(O) ) mac ac T(l) mec ac e,as ee e,as' e,as •
(3-96)
In third order of E the results of section 1 change as fdllows.
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-1 one obtains:
af(O) 2C kT(O) aln(f(O) ) _e,as = ~ L{f(O) (c) [n(O) (1+ ~ e,as ) + OTz 2 ac e,as e,as m c ac
·~ e
aln(f(O) ) ~ ac e,as )dv)} +
(3-97)
Page 42
-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 fA:
o, (3-89)
where the constant of integration has vanished by consideration
of the limit c + oo. The equation for f A can be written in the
following form:
where B(w)
v2 Te m e
o,
m c 2 e
w '" 2kT ' "ee A m v3
e Te
The following normalizations should then be imposed on a
solution of equation (3-99):
QO ~ . ;;
Jexp(y)w dw = -z , 0
QO 3/; Jexp(y)w3 12dw = ~ , 0
(3-99)
(3-99a)
(3-100)
in order to determine the integration constant and the
temperature TA. If w>>l, then the solution of (3-99) may be
approximated by the solution of the following equation:
* T d d B(w) (1+ Ta !!I.d ) + v (1+ !!I.d ) .. O.
A w ee w (3-101)
The solution of this first order differential equation is:
Page 43
-36-
w B(w')+v - J( * ee }dw' + C,
o B(w' )T /TA +' v a ee
y(w) (3-102)
where the integration constant C and the temperature TA are
fixed by conditions (3-100).
The problem has' been investigated earlier by Lo Surdo 9, 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.
Page 44
-37-
References
1. w. van de Water, Physica 85C(l977)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 1.Q.(1930)61,.!_(1934)1003.
8. Braglia et al, Il nuovo cimento 62B(l981)139.
9. C.Lo Surdo, 11 nuovo cimento 52B(l967)429.
Page 45
-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£ (cf. equation (2-23)).
This means that the degree of ionization 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 paperl. 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 £ leads to some
results which are not found with the usual harmonic tensor
expansion 5. •
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. Exprei.sions for the electron transport coefficients
will be derived and finally the modifications in case of a
seeded plasma are given.
Page 46
-39-
IV-1 Heavy particle results
The heavy particle Boltzmann 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 g and the
multiple time scales formalism (MTS) is applied. 'up to second
order the results are:
(4-1)
(4-2)
(4•3)
o, (4-4)
(4-5)
(4-6)
By means of an H-theorem obtainable from equation (4-1) it
follows that f~O) relaxes to a Maxwellian when T0
+ w. This
limit will be indicated by a subscript "as" so that:
m m Iv - w(O) 12 = n(O) ( a )3
'2exp{- a - -a,as }.
a,as 21rkT(O) 2kT(O) f(O) a,as
a,as a,as
In order to proceed the moment equations are needed. The
balance equations fo-r the neutral particles read:
(4-7)
Page 47
-40-
Cln _a + £ 2V•(n w ) = 0 Clt a-a '
(4-8)
Clw m n (-=..a + £2(w •V)w )
a a Clt -a -a (4-9)
ac n (-a+
a Clt £2w •Vt)+ £2V•(n + P •w ) = £ 2!\m v2J (f f.)d3v
-a a ~a =a -a a ai a' i '
(4-10)
in which the interaction terms between the heavy particles and
the electrons are omitted because these are of the order £4 •
The macroscopic variables are also expanded in powers of £ and
the MTS formalism is exploited. Up to second order the results
from these equations are:
Cln (O) Cl~(O) Clw (O) a a -a
o, ho =ho = ho (4-11)
Cln (O) Cln (l) Clw(O) Clw(l) at(O) ae < 1) a a -a -a a a
o, h 1 = ho = hl =ho = hl =ho (4-12)
Cln(O) Cln (l) Cln( 2) + V•(n(O)w(O» a + a + a o, aTz h 1 ho a -a (4-13)
(4-14)
(4-15)
From equations (4-7) and (4-12) and the definition (2-4) of
chapter II it is concluded that 4- f;o)= O. Then equation (4-2) a"( 1
becomes in the limit -r 0+ 00 , indicated by a subscript "as":
J (f(O) f(l) ) + J (f(l) f(O) ) = O. aa a,as' a,as aa a,as' a,as (4-16)
This equation possesses the following general solution2:
Page 48
-41-
f(l) a,as
= (a + a •v + a v2)f(O) l -2 - 3 a,as' (4-17)
where ai(!,< 1,T 2, ••• ) are at this point arbitrary functions of
position and time. The Chapman-Enskog choice:
n(l) = w{l) = T(l) = O, a,as -a,as a,as
(4-18)
makes these functions zero, so that the first order correction.
to f(O) vanishes: a,as
f(l) = o. a,as {4-19)
Next equation (4-5) will be considered in the limit , 0+ <»:
of(O) i,as = J. (f~O) ,f(O) ).
at1 ia i,as a,as (4-20)
This equation also possesses an II-theorem implying that t(O) i,as
relaxes to a Maxwell distribution function, when ,0
+ oo, with a
temperature and a hydrodynamic velocity equal to the neutral
ones:
(0) f (r,v,,2
, •• ) iA - -
(4-21)
A subscript "A" denotes the limit Tl+ oo, The ion balance
equations read:
ani e;2\'-(n w.) = o, (4-22) - + at i-i
a!'i e: 2(w. •V)w ) e;2V•P .- e: 2en E - e;2min.w .wixb = mini (at + + -i -i =i i- i ci- -
(4-23)
ati ni{at + e;2!1•Vti} + e;2V•(gi+ ~i·~i) - e;2eni§·~1= e:J\miv2Jiad3v
(4-24)
After expansion in powers of e: and using the MTS formalism the
results up to second order of e: are:
anio) a!io) aTio)
ato = ato = ato "' o, (4-25)
Page 49
-42-
(4-26)
(4-27)
(4-28)
a (O) a (1) a (2) ni ni ni · (0) (0)
- + - + ~ + 'J•(ni :!!1 ) = O, OT2 OTl OTO
(4-29)
= /m v(J. (f(O) f(l)) + J (f(l) f(O)))d3v i- 1a i • a ia i • a • (4-30)
(4-31)
(1) (1) Furthermore the first order corre.ctions n1A and TiA are
assumed to have vanished. Equations (4-29) - (4-31) then read: a (O) niA + n ( (O) (0)) 0 iT
2 '• niA l!!1A = • 1 (4-32)
(0)
m n(O){a!aA + (w(O)•V)w(O)} + 'Jp(O)_ en(O)E' i iA oT 2 -aA aA iA iA -
(4-33)
Page 50
-43-
(4-34)
(4-35)
Note that t(O) = i(O) = ~T(O)+ ~m lw(O)l 2 if mi/ma =b(I).
iA aA 2 aA a -aA With the definition of the total derivative:
~=.L+w(O).v dt 2 ct 2 -aA '
equations (4-32) and (4-34) can be written as follows:
d (0) niA + n(O)V•w(O)
a:[2
iA -aA o,
And for the neutrals the Euler equations are obtained:
(0) dnaA + n(O)V•w(O) dt 2 aA -aA o,
(4-36)
(4-"3h (
"·' (4-38)
(4-40)
dw{O)
man~~)~:A + Vp~) = O, (4-41)
(0) 3 (0) dTaA (0) (0) -zr1aA k0t2 + PaA V•lfaA ., O. (4-42)
When equations (4-39) and (4-42) are compared with each other
it appears that there is no net energy exchange on the t 2-time
scale between ions and neutrals in first order:
I\m jv - w(O)l2J (f{l) f(O))d3v = O
i - -aA ia iA ' aA '
which is compatible with the choice T(l). o. iA
(4-43)
Page 51
-44-
Now equations (4-3) and (4-6) can be treated. When T0+ ~ these
equations read as follows:
(4-44)
()f(O) . (0) eg' (0) 1f'T~A + y•Vfi~)+ (~aA -y)•----zoy fiA = Jia(f~~),f~~)),
kTaA (4-45)
which are the Chapman-Enskog equations for f~~) and fi~)· The
left-hand sides of these equations can be brought into a more
familiar form through a transformation in velocity space:
! + y - ):!~~) from the laboratory frame to a frame moving with
the zeroth order hydrodynamic velocity of the neutrals.
With the aid of the macroscopic equations (4-37) - (4-42) the
source terms of equations (4-44) and (4-45) become:
<lf(O) aA + "f(O) 1fT y•v aA • 2
2 m c 5 (0)
H~ny '2")s •Vln(TaA ) 2kTaA
mi 5 (0) c•{(----)Vln(T ) + - 2kT(O) 2 aA
aA
(4-47) (0)
where £ = y - ~aA , 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 ni<< na the equations (4-44) and (4-45) are recovered
with source terms (4-46) and (4-47) respectively. The solution
of these equations is standard 2• If resonant charge transfer
instead of elastic scattering is the main mechanism for the
Page 52
-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.
Page 53
-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 £
with. Equation (2-17} then reads:
v - w will be started -a
af ate+€(£+ €!a}•Vfe- ooce~·(~xVcfe} +
eE aw - {~ + eoo w xb + ~ta+ e2 ((c + ew )aV)w }•V f m ce-a - a - -a -a c e
e
• J (f ,f ) + eJ (f ,f ) + eJei(fe,fi}, ea e a ee e e
where ~ is a unit vector in the direction of ~ and
the electron cyclotron frequency. The hydrodynamic
(4-48}
oo •~is ce m e
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
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:
af(O) e
a.to (4-49)
(4-50)
Page 54
-47-
(4-51)
From equation (4-49) an H-theorem can be derived, see also
chapter III,:
(4-52)
where (4-53)
so that again the zeroth order distribution function relaxes to
an isotropic function when t 0+ ~. In that limit one obtains
from (4.50):
af<0> eE' af(O) ~e,as + r•(Vf(O) - -=-~e,as) - w (bxV f(l) )•c = at 1 ~ e,as mec ac ce - c e,as -
J(O)(f(l) ) + J (f(O) f(O) ) (4-54) ea e,as ee e,as• e,as '
where Je(Oi)(f(O) ) vanished because of the isotropy of f(O) • 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:
of(O) ~e,as = J (f(O) f(O) ) at
1 ee e,as• e,as •
w c•(bxV f(l) ) + J(O)(f(l) ) = c•A'f(O) ce- - c e,as ea e,as - - e,as
(4-55)
(4-56)
where: JI.' e~· a
V - iii"'C'ac, as in chapter III. Equation (4-55) e
Page 55
-48-
also permits an H-theorem which states that f(O) relaxes to a e,as
local Maxwell distribution function when T1+ ~:
2 (0) (0) me 3/2 mec
feA = neA ( (0)) exp{- (O)}• (4-57) 21rkT eA 2kT eA
A solution of equation (4-56) can easily be obtained if f{O) e,as
is developed into harmonic tensors (see appendix C):
f(l) (c) = f(l) (c) + f(l) (c)•c + f(l) (c):<cc> + e,as - e,as -e,as - =e,as --
~ f(l) (c)•<c~ l n-e as n - '
n=O ' (4-58)
where f(l) is a tensor of rank n and • denotes an n-fold dot n-e,as n product.Insertion of this expansion into equation (4-56) gives:
I {( 1 - nw bx) f(l) (c) }-<c~ n=l T(l)(c) ce- n-e,as n -
= - c•rll' f(O) - - e,as' (4-59)
1 11 where : --(-) = 2.11n cf o(c, x) (1-P (cosx) )sinxdx,
T(l) c a 0 n (4-60)
(see appendix A).
If b is directed along the z-axis ~(n) in index notation reads:
M(n)ij = oij - nwceT(n)~c)eikjbk, (4-6la)
M-1 =(n)ij (4-6lb)
Only the first two terms in the expansion of f(l) e,as are non-zero
so that the solution of (4-59) is:
(4-62)
where f(l) is a yet undetermined isotropic contribution. It is e,as in fact th!!! homogeneous part of equation (4-56). In much the
same way equation (4-51) will be treated. When T0+ ~ the
isotropic part of this equation reads:
Page 56
df(O) af'(l) e,as + e,as
<l"12 "'fr 1
111 c 2 a
-49-
+ J (f(O) f(l) ) + ee e,as' e,as
+ J (f(l) f(O) ). ee e,as' e,as (4-63)
This is the Chapman-Enskog-like equation determining the first
order isotropic correction. The non-isotropic parts give the
following solution for f( 2) in the same way as in the case of e,as
the first order part:
f( 2) (c) e,as
f(2) -e,as
1(2) (c) + c•f(2 ) (c) + <cc>:f( 2 ) e,as - -e,as -- =e,as
2C n(Q) ei i,as f(l) m c3 -e,as
e
2C n(O) u(O) af(O) ei i,as-i,as _e,as +
111 c 4 e
+ J (f(l) >} 1 ee -e,as '
<k
af(O) f(2) = T M-1 •{- ..'t'f(l) _ .!. _e,asVw(O) } =e,as (2)=(2) - -e,as c ac -a,as '
where the constant C is defined in appendix A. ei
(4-64)
(4-64a)
( 4-64b)
Again there appears a yet undetermined isotropic contribution.
In equation (4-64a) the following linearized electron-electron
collisio.n term was introduced:
J (f(l) ) := ee e,as
J (f(O) f(l) ) + J (f(l) f(O) ) ee e,as' e,as ee e,as' •,as
= J a<l) ) + c• J (f(l) ) + ••• ee e,as - 1 ee -e,as (4-65)
This expansion is justified because the collision operator is
rotationally invariant in velocity space.
In expression (4-64a) for the correction f(Z) the contribution -e,as
of the first order isotropic correction appears. The last two
terms between braces express the influence of the Coulomb
Page 57
-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 higheT order terms. In appendix E it is
shown that one can replace mec3 by [mec3 + 2Ceini~~s'(I)(c}] in
the denomerator of that specific term. 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 1+ 00 , it has been demon-(0)
strated that feA is a Maxwellian and a solution of (4-63) can
be constructed. Before doing so the electron balance equations
will be dealt with first.
Page 58
-51-
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 £ • v - one can see that:
1 ~ fcf (r,v,t)d3c = w - w = u • n - e - - -e -a -e (4-66)
e
The electron balance equations for a WIG provided with the
appropriate powers of e: then read as follows:
an -;;-e + e:V•(n u ) + E2V•(n w ) = O, ot e-e e-a
au m n {~-e+ e:(u •V)u + e: 2(w •V)u } +
e e at -e -e -a -e
aw
e:en E + e-
(4-67)
+ e:mene(a~a + e:(~e·V)!a+ e: 2(!a•V)!a) + menewce<~e+ e:!a)x~
= fm v(J (f ,f) + cJ .(f ,fi))d3c, ·e- ea e a ei e (4-68)
ai ne(-;;-te + eu •VC + e:2w •vi ) + e:V•(n +
o -e e -a e ~e
aw + e:en u •E + em n u •(.,-a+ e:2w •Vw ) + E2(p + m n u u ):Vw = e-e - e e-e at -a -a =e e e-e-e -a
= f\m c2(J (f ,f) + e:J 1(f ,f1))d3c. e ea e a e e
Where now, slightly different from equation (2-4):
l, = ~23 T + L- u2 • e e '2'lle e
(4-68)
(4-70)
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:
an(O) e
'a:r 0 o, (4-71)
(4-72)
Page 59
-52-
And in first order of E:
(4-74)
(4-75)
Here and in the sequel the results are used that were obtained
in preceding sections, e.g. i- w(O)=O. When To+ oo, equations To-a (O)
(4-73)-(4-75) can be further simplified, because then f is ) ( ) e, as
isotropic, which implies: u(O = 3 O 0 etc. In this limit , -e,as e,as the first order equations become:
(4-76)
f(l)
m c + m n(O) w (u(l) + w(O) )xb + J e- e,asd3c =O.
e e as ce -e,as -a,as - T (c) V (O) + en (O) Pe, as e
(l) (4-77)
Substituting the expression for f{l) as given in equation e,as
{4-62) obviously renders equation (4-77) into an identity.
Further obs6rvation shows that equation (4-77) 'closes' when
T(l)(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-i+.
Equation (4-77) in case of Maxwell interaction reads:
(0) m n
Vp(O) + en(O) E + m n w (u(l) + w(O) )xb + e e,asu(l) =O, e,as e,as- e e,as ce -e,as -e,as - T(l) -e,as
(4-78)
Page 60
-53-
which is the generalized law of Ohm in first order.
In second order the balance equations will be considered
asymptotically when T0+ 00 , then they read as follows:
lln(O) lln(l) _e,as + e,as + 3T 2 hl
'V•(n(O) (u(l) + w(O) ) ) e,as -e,as -a,as o,
3u(l) m n(O) {_:_e,as + w u(Z) xb} + V•P(l) + en(l) E' =
e e,as ClT 1 ce-e,as - =e,as e,as-
m c c J....!:.:.... f( 2) d3c - 2C n~O) J-=- f(l) d3c, T(l) e,as ei 1,as c 3 e,as
(4-79)
(4-80)
dT(O) <lT(l) 3 k{ e,as + _e,as + 11 (1) •VT(O) } + en(O) u{l) •E' + "! ne,as 'd'T
2 <lT
1 =e,as e,as e,as-e,as -
+ 'V•(n(l) + P(O) •u(l) ) + P(O) :vw(O) ~e,as =e,as -e,as =e,as -a,as
m m c 2 kT(O) = - ~f~ (1+ ~ !.....)f(O) d3c. (4-81)
ma T(l) mec Cle e,as
d The derivative -;r; was defined in equation (4-36) of section 1. "2
Now the following Ansaz is made:
n(l) e,as
= T(l) = O, e,as
which will be verified in the next section.
(4-82)
The equations (4-79)-(4-81) then assume the following form when
T 1 + "" (subscript A):
d (0) 0
eA + n(O)v•w + V•(n(O)u(l)) = O, T. 2 eA -aA eA -eA
+ (0) 1 (0) peA ·~aA
m T(O) = ~(...!! -
m T(O) a eA
(4-83)
(4-85)
Page 61
-54-
Next the local entropy density in zeroth order is introduced:
(4-86)
Then it is possible to rewrite equation (4-85) as an entropy
balance equation:
(4-87)
where the thermodynamic forces:
kT(O)
X' e {~· + ~ Vln( (O»} (4-88a) :== -m m e PeA
e
x := -'Vln(T(O» (4-88b) -q eA
have been introduced. The first term 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(t)' It also gives the
relations between the fluxes i(Al)= m n<6A u(Al) , i(Al) and the -e e e -e e
forces as defined in (4-88a,b). These relations, which also
obey the Onsager reciprocity relations, read as follows:
i (1) -eA
(1) .9eA
(4-89a)
(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=(l) 1 f ( ) 2f(O)M-l d3
= ---coT '(l) c c eA =(1) c, 3neA
(4-90a)
(4-9Pb)
Page 62
-55-
kT(O) m c2 '(1) = _:A f ( ) 2(_e_ - 2)2f(O)M-1 d3 ~ 3 T(l) c c (0) 2 eA =(l) c.
2kTeA {4-90c)
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 ,1+ ~will be
given. Again an Ansatz is made:
n( 2 ) = T( 2 ) = 0 eA eA ' (4-91)
which will be verified later on. In third order of E there
results from equations (4-67)-(4-69) when <1+ ©:
a (O) neA + V•(n~~\1~!)> = O,
h3 (1)
(0) {d.!!eA (1) (1)} meneA a:;:- + YeA • 17»eA + Tz
(4-92)
m n(O)w (u(3)+ w( 2))xb = -Jme£ f( 3)d3c - 2C n(O)J ~ f(Z)d3c + e eA ce -eA -aA - T(l) eA el iA c3 eA
m2 . + -8~m n(O)v u(l) + -~{v-2 A•X' + B•X)
312if e iA ei-iA ma Te = -m = -q ' (4-93)
(4-94)
The tensors ! and ~ in equation (4-93) are defined as follows:
c 2 (0) -1 3 ~ := Ja(c>) T(l)(c)feA ~(l)d c,
m c 2 c 2 ( e 5) (0) -1 3
~ := Ja(c>j' '(l)(c) ZkT(O) - 2 feA ~(l)d c, eA
(4-95)
Page 63
where: a(c) = -1-
T(l)
-56-
kT(O) aA a - -- -(cl+
2ci+ ac l.. ). ac \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:
(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 product.ion 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(l) 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 T(l) the cross effects are absent in first order.
Page 64
-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 T0+ ~
is a Maxwell distribution function. The equation for the
isotropic correction f(~) then reads: df(O) e m c 2
J cf(l)) = _eA + .£\,.f(l) + e f(O)V•w(O) + ee eA dT 2 3 -eA 3kT~~) eA -aA
where:
+ m 3 T
+ e a [ c r a 1 )f(O) ] m czac-,- l("O) - eA '
a (1) TeA
2 (0)
(4-97)
T(o)- ema'(I)TeA 1 T+ = - E' •M- •{v-2 X' +
a aA 3m2 - =(1) Te -m
m c2
(~(0)- f )! }. (4-98) 2kT q
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) = Jee(f f(O)) + J (f(O) f) ' eA ee eA ' ' (4-99)
i.e. asymptotically on the t1
time scale. The moment equations
(4-83) and (4-85) will be used to eliminate the time derivative
in the first term 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:
4 c p+2 I (f) = ~ Jv f(v)dv,
P cp o (4-100)
it is possible to integrate equation (4-97) once. The
integration constant vanishes, as can readily be verified. The
result is then:
Page 65
-58-
(4-101)
It is then advantageous to make a change of variables from
(c,_!:,< 2) to (W,_!'.;< 2) where:
m c2 e
w :=-CO) , 2kTeA
(4-102)
~he functional notations are not altered after this
transformation. Further the function g is introduced which is
related to f~~) according to:
g(w) := {l+ .!_)7(l). aw eA (4-103)
Finally an integral equation for the function g is obtained:
lw w312~ F(w)g{w) - 3Jx312g(x)dx - ~3~ Jg(x)dx = b(w), (4-104)
0 w
in which the source term is defined by:
-b(w) =-'-[-G (w)-2....E'•M":"l •X' - (G,(w)-.fG
25
1{w))e_E'•M:,.(l1)•X_q+ 24nC 1 2 - =(l) -m v ee vTe ,
2 {O) 3m TaA 1 (0) (O)
+ -;2 G2(w) ((o)-1) + (o) V• (meneA ~s(w) ·~~+ P eA ~6(w) ·~q) mai: TeA neA
51{1>.x ( 1 2 1 ) l w312 [ -eA -q + + m G3(w)M:(l) •X'+ vT G4 (w)M-(l) •X •X - ----...,.,,... -
e = -m e = -q -q 24nC n(O) 2
eE' •i(l) - -eA
kT(O) eA
ee eA
4m2 ~ e J x312exp(-x) J
+ -- T (x) dx , m/n 0 (1) (4-105)
Page 66
! where: '(l)(w) =
Q(w)iw'
-59-
! ..
ew ""I {2./X s x 1 } -x G5(w) = w /; =0 - Q(x)~l) e dx,
5 w J00
r2/X R x(x-2) 1 } -x G6(w) = e l r- .. o - Q(x) ~l) e dx,
w l'1T
00 -x _ f xe 1 ~o - Q(x) ~l)dx,
0
5 -x oo x(x- -)e
~o .. J Q(x)2 ~({)dx, 0
The function F(w) is defined as follows:
F(w) := 1T\ewerf(w\)/4 - w\/2 •
of which some properties are:
(4-106)
(4-107)
(4-108)
(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:
(4-110)
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 s·imple ordinary
differential equation for the isotropic correction:
d"' d ~(l) (dw2 + 2dw + l)feA = J(w),
where the new source term J(w) is connected with b(w) in
equation (4-104) via the relation:
(4-111)
d 1 d w w -x d ew w -x d J(w) = dw[F(w) Tw{e fe b(x)dx}] = dw[F(w)fe tic"(x)dx].
0 0 (4-112)
Page 67
-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=w312, 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~i) reads:
7(l) = e-w ff exJ(x)dxdw' + c1e-w + c2we-w eA w w'
(4-113)
The constants c1 and c2 are fixed through the requirements in
(4-82) leading to: 00
J f(l)wl/2dw = 00
f f(l)w31 2dw = O. eA eA (4-114)
0 0
Thus it was legitimate to make the Ansatz (4-82).
Again in the special case of Maxwell interaction between
electrons and atoms (•(l)= constant, i.e. Q(w) proportional to
w\) the source term b reduces to:
m w512
b(w) = 60~Ce (O)({<t - w)!q + (Vln(p~~)) - v)}•g~~>j. (4-115) eepeA
Only those parts relevant for the solution are given here. This
expression vanishes for a homogeneous electron temperature.
Page 68
-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 "!f(f) becomes equal
to the unit tensor. As a consequence the tensors ~ 5 and ~6 , ~o•
~O become also proportional to the unit tensor so that e.g.
• G 5~. The electron fluxes in first order then read:
i(l) -eA
:= fm cf(l)d3c = m n(O)D {v-2 S X +RX } e- eA e eA 0 Te O:.:m 0-q • (4-116a)
Ao{vT; Ro~m + Lo~q},
(4-116b)
4TkT(O) eA
4:fn(O)(kT(0))2
eA eA 5 2 -w "'w(w--) e
Lo • J Q~w) dw. m 3/iT
e
AO = _ _.;;.;;;;.___...;...;.;-..._ lll 3/i[
e 0 ( 4-117)
Observe that n(l) JeA 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( 2 ) = fm cf(2 )d3c = m n< 0 >n 0[sv- 2(-s + !.. s )x + -eA e- eA e eA Te ei Ii[ ee -m
4 . 2fTI eE s(R - - R )x + .L.!(- c - _-_ B+ + B3X_q) ].
ei ,- ee -q 128 -1 (0) 2 vn kTeA
(4-118)
m c 2
~e(A2 ) = J....!L... cf(2 )d3c - ~(O)u( 2 ) = A0[sv-2(-R + !.. R )x 2 - eA 2 eA -eA Te ei /Ti ee -m
S(L - !__ L )x + fl21TI(- C
ei ;-; ee -q 128 -4
± where: B1
=
(4-119)
-1)
k X •X + 31-m -q + k x2.
2 4i q vTe
(4-120)
Page 69
-62-
In the expressions (4-118),(4-119) the following coefficients
were introduced:
s ee R ee
00 -w .. f f(w)we dw ,
Iei(f) := o Q(w) {w2q(w)+B}
Iee(f,g) = ff(w)Z{g(w)}dw. (4-121) 0
where .:t, 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;
(0) (0) - 3me TaA
Vln(naA ) )Bi + k2im 12 v\(O)) + { ( eE )2 2 kli -(O) - k4ix
kT q a eA eA
+ (k - k )X •X - k (V•X - x2) - k V•X }x + 6i 3i -q -p Si -p -p 6i -q -q
( e ,2 2 2 eV(V•E) + k11 l-(o)J VE + k4i vx - kSi vx2 - kSi co) + kSi V(V•lfp) +
kT q p kT eA eA
+ (k3i-k6i)V(!q•lfp),
where: X := -Vln(peA(O)). -p
A mean free pathlenght i has been defined by:
v 2 = kT(O)/m. Te eA e
(4-122)
(4-122a)
(4-123)
The parameter B is of the order e and is proportional to the
e-e to e-a collision frequency ratio:
B :• v "T/2 , ee
v ee
n< 0>c eA ee
2m v3 e Te
c ee e 4 lnA D---. 8H2m o e
(4-124)
Page 70
-63-
The coefficients kij appearing in the expressions above are
defined as follows:
.. .. "" kij := fe-w{ f f exJ1(x)dxdw' + Cli + c2iw}Hj(w)dw
o w w'
The functions Hj are defined by:
H2(w) = dHl dw
H (w) = w(w - 5/2) 4 Q(w) '
H (w) = ~ 5 dw '
(4-125)
(4-126)
(4-127)
Note that in general the coefficients kij 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 E = 0
2 x = -\7ln(p(O)) -p eA
3 x = -Vln(T(O)) -q eA
4 i{l) = 0 + eE -eA -
5 (1) = 0 + eE .9eA -
= 0
= 0
= kT{O)(~X eA -q
= kT(O) (k X eA >.-q
+ x ), -p ~
+ X ), k;. -p
= Ro/So
Lo/Ro 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:
Page 71
(2) .9eA
-64-
(4-128a)
(4-128b)
In these equations the second order fluxes i(Z) and n( 2) are -ex .. ex different from the other terms because they are not
proportional to E_, X or X • They give no corrections to the -q -p
first order fluxes, but are new effects. Their general fol'lll is:
i (2) - a,2{; (O)D 9l ---mn 0 aY -ex 126 e eA i=l i-i '
in which the vectors !i are defined as follows:
!1 ( e )2 2 !z = _e_ V(V•E), !3 = _e_ V(E•X ) kT(O) VE ' kT(O) - kT(O) - -p '
eA eA eA
y - vx2 V(V•X ), !G e
-4 q' -p --CO> V(Jj!·~q)' kTeA
Y7 = VX2, y ,. V(V•X ), y = V(X •X ). (4-130) - p -8 -q -9 -p -q
The coefficients ai and bi can be expressed in terms of kij:
al = kll' bl = kl4'
a2 = k4 l' b2 k44'
a3 = -a4 -as ks 1 • b3 -b4 = b5 = k54'
a6 = k61' b6 k64'
a7 = kll - ksl' b7 = kl4 - k54' 5
- 2k31 - ksl' bs 5
- 2k34 - ksi+' as = ¥<-11 = ¥<-1'+
a9 = k31 - k61' b9 = k3'+ - 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
Page 72
-65-.
temperature gradient is present, reads:
il) + i2)
where: 2 (0)-
4 e neA t (J = -o 3v'ir me
(4-132)
(4-132a)
The thermal heat conductivity in case 4, where there is no
first order electrical current, is:
A (l) + A <2> = A {L - k R + .eh'ii[K X2 - K2X •X r K3V'•X + 0 0 T 0 12$ l q -p -q -q
+ ~L - k R ) - 0 (L k R ) } fi' ee ---r ee '"' ei - T el •
where:
Kl = Y16 - kTY15 + \Y14 + kik14 - k44'
Kz = Y26 - kTY2s + \Y24 + k34 - ZkTkl4 - k64'
K = 3
K4 = k26 - kTk25 - \k24'
5 yli = k4i - ZkTk3i +2i<-Tkli + kikli'
Yzi = k61 - kTkSi + k3i - ~li - kTkli'
Y31 = k6i - kTkSi'
(4-133)
(4-133a)
The thermal diffusion coefficient for the electrons up to
second order in case 4 reads:
Page 73
D(l)+ D(Z) = D {R + (~k T T 0 0 21
-66-.
(SR + _il R + ei r; ee
R, 2f1i + 12"i3[-Ks!q•!p + K6V•!q + K.f~ - k51V•_!p + (k51 + kll)x;]}.
where: Yz3 - ~Y21 + k31 - k61 - ZkTkll'
Y33 - ~Y31 - k61'
Y13 - ~yll - k41 + kik11•
(4-134)
(4-134a)
Finally, again in case 4, the coefficient for the Dufour effect
for the electrons reads into second order reads:
+ R,22.r;[Y15x2 - Yzsx •X + Y35V•X ]}. i a q -q -p -q
{4-135)
All the foregoing results for case 4 can be transformed to
those of case 5 by simply replacing~ by kA.
Further it is observed that only those parts in expression
(4-134) and (4-135) that originate from the isotropic
correction do not obey the Onsager symmetry relation.
Numerical examples are worked out in chapter VI.
Page 74
-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
tion 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 Jeb(fe,fb) is added where b
denotes the seed. In the expressions for f(l) one has to -eA s
replace T(l) by T(l) defined as:
Inserting the expressions for <~i~ one obtains (see (4-106)):
s T(l)
-a T
--------- =: -a IW{Qa(w)+~b(w)}
T
(4-137)
11here an electron-atom cross section for the seeded plasma has
been introduced. When the seed has the same temperature as the -a -b
neutral gas it follows that T /T m nb/na i.e. proportional to
the.relative seed concentration.
The collision terms in second order of e 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:
( 2) (2) me a c3 J (f) + J (f) m -,,,.,,.-.{- f + ea eb m c'ac T
a s,m
where:
kT(O) ~ _£!_1.f_}
m , Tac ' e s, (4-138)
Page 75
T s,m 1 m 1 -1
(-·- + ....!__) a m.. b '
'(1) D '(l)
-68-
's, T (4-139)
If the heavy particles are in thermal equilibrium these
collision times are equal. The first term in the right-hand
side of equation (4-63) should be replaced by expression
(4-138) with f = f(O) • In short the modifications to be made e,as for a seeded plasma are: replace evrywhere '(l) by expression
(4-136) except in the energy equation: (4-81),(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:
* m 3 T e a [ c ( a 1 )f(O)] m c:zac-,---zof - eA '
a s,m TeA
where now:
em -r 8 T T(O)
a (1) s,m eA E' •M":l •{v-2 X' + 3m2 - =(l) Te -m
e
(4-140)
m c 2
(~(O) - t):f }. 2kT q
eA (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.
Page 76
-69-
References
1. w. van de Water, Physica 85C(l977)377.
2. $.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, 11 nuovo cimento
53B(l979)95.
5. I.P.Shkarofsky,T.W.Johnston and M.P.Bachynski, "The
particle kinet.ics of plasmas", Addison Wesley, 1966.
6. P .M.Banks and G .J. Lewak, Phys. fluids !!_( 1968)804.
Page 77
-70-
V STRONGLY IONIZED GASES
A strongly ionized gas was defined in chapter II 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
Ramsauer minimuml. 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 Landshof 3 and Kaneko4 among others. With the
inclusion of a neutral species the problem has been attacked by
many authorsS-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:
Page 78
-71-
n Q(l) e ee
n Q(l) a ea
= 0(1). (5-1)
The electron-electron collision cross section Q(l) is much ee larger than Q(l): the electron-atom cross section. Therefore
ea the assumption will be made that the degree of ionization is of
the order e::
n e
n a
= (J{e:), (5-2)
which, of course, implies a limitation for the validity of the
heavy particle equations. With the assumption (5-2) the heavy
particle Boltzmann equations read:
(jf _a+ e:2v•'Vf = e:3J + J + e:J (5-3) at - a ae aa ai'
The right-hand side of equation {5-4) contains extra .factors £
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
e2 because of the inefficient momentum transfer process (cf.
equations {2-24) and (2-25) of chapter II).
The electron kinetic equation now reads:
af eE aw _e + ec•Vf + e2w •'Vf - ( e--=. + ew w xb + £..:_a +e2(c•'V)w + at - e -a e ' m ce-a - at - -a e
+ i;;3(w •V)w )•V f - w c•(bxV f ) = J + J + J i' -a -a c e ce- - c e ea ee e (5-5)
where the transformation to the hydrodynamical velocity of the
neutral gas has been made as in chapter IV:
c :• v - w • - - -a
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.
Page 79
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V-1 Heavy particle results
The heavy particle equations are only slightly altered
when compared with the weakly ionized gas in chapter IV sect.!.
Equations (4-1) and (4-4) remain unc~anged so that f(O) is a a,as
Maxwellian as in (4-7). The equation of continuity for the
atoms is also identical to (4-8). The factor i:::2 in the right
band sides of (4-9) and (4-10) is now replaced by i:::. The
results in zeroth order from the balance equations are again:
= o. {5-7)
The results from the continuity equations are the same as in
the case of a WIG. Up to second order they read:
(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:
aw(O) aw(l) m n(O){...:::_a +-=._a } = fm vJ .(f(O) f(O))d3v
a a 3T 1 3T 0 a- ai a • i • (5-9)
(5-10)
(5-11)
Page 80
-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 c:
af(O) af(l) i + ~i = J (f(O) f(O)) + J (f(O) f(O)).
a:rl a,o ia i ' a ii i ' i
It is shown in appendix B that from these equations an (0)
H-theorem can be derived implying that f 1 relaxes to a
(5-14)
Maxwellian, with a hydrodynamical velocity and a temperature
equal to those of the atoms, when 11+ oo:
m m Iv - w(0)12 _ (0) ( i )312 { i - -a } - n ( ) exp - ( ) •
iA 2nkTa~ 2kT~ (5-15)
Contrary to the case of a IHG the conclusion that f(O) does a,as
not depend Qn 1 1 cannot be drawn. From equation (5-13) it
appears that only when 11+ m the same equatTon for the first
(1) order contribution f aA as in the case of the WIG is obtained:
With the Chapman-Enskog choice:
n(l) = w(l) • T(l) = 0 aA -aA aA •
(5-16)
(5-17)
it can be concluded that the first order correction is absent
if "1+ oo:
f (l) = 0 aA - • (5-18)
The second order equations derived from (5-3) and (5-4) read:
af<0> af< 1> af< 2> ~a + a +~a + v•Vf(O) = J (f(O) f(2)) + J (f(2) f(O)) oT 2 lit 1 aT 0 a aa a • a aa a ' a
(5-19)
Page 81
-74-
When T 1+ ~these equations reduce to the Chapman-Enskog (1) (2)
equations for the corrections fiA and faA :
(5-21)
(5-22)
In order to evaluate the right-hand sides of these equations
the balance equations have to be considered in the limit T1+ ~.
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)
(O)d!aA (0) (0) (1) (O) 3 miniA C'fT
2 + VpiA - eniA ~· = !miyJia(fiA faA )d v, (5-24)
- p(O).!!._ln{n(O)(T(0))_3/2} = f~m Iv - w(O)l2J. (f~l) f(O))d3v.
iA d1 2 iA aA i - -aA ia iA ' aA
(5-25)
The macroscopic atom equations read:
(5-26)
(5-27)
- (O).!!._l { (O)(T(0))-312} = !Lm Iv - w(O)l2J (f(O) f(l))d3v, PaA d1 2° 0
aA aA ~ a - -aA ai aA ' iA (5-28)
Page 82
-75-
Addition of (5-25) and (5-28) gives with the aid of (5-23) and
(5-26):
i._ln{n(O)(T(0))-312} = ~ln{n~O)(T(0))-312} = O. dt 2 aA aA dt2 1a aA (5-29)
From (5-25) and (5-28) it can then be concluded that there is
no energy exchange between ions and neutrals on the t 2-time
scale in this order:
(5-30)
where £ = ! - ~~~), 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 t 2-timescale. Addition of the equations (5-24) and
(5-27) gives:
o, (5-31)
(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(O) one finally obtains:-
- -aA
(5-33)
Page 83
-76-
(S-34)
where nh = n~)+ ni~) and the diffusion driving forces are
defined by:
d = -d = -ia -ai
(0) niA
v(-) -~
m n(O)n(O) aiA aA E'. phkTh~
(S-35)
These equations can be seen as a special case of the ones
obtained by Chmieleski and FerzigerB. This is due to the
restriction made in relation (5-2). Their equations for the
heavy particles are coupled, whereas here equation (S-34) can
be solved independently for ril)· Substitution of the solution
into (5-33) then gives an equation for f~)· The solutions can
be obtained by means of a traditional Sdnine polynomial
expansion 10 •
The ion- and atom Chapman-Enskog equations (5-33) and (S-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.
Page 84
-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-theorem can be
derived implying that the zeroth order electron distributton
function relaxes to a local Maxwellian when 'a+ oo;
f(O) m 3/2 m c 2
n (0) ( e ) exp{- e }. (5-37) e,as e,as 211kT(O) 2kT(O) e,as 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:
(5-38)
When i: 0+ oo the first order equation reads:
<lf(O) e,as
a:rl eE'
+ c•(V +-:-=TO)- )f(O) - e as - w (bxV f(l) ) •c ce - c e,as -
/O)(f(l) ) + ea e,as
kT ' e,as
(5-39)
where Jee(f) is the linearized collision operator (see (4-65)).
In the next section it is shown that n(O) and T(O) do not e,as e,as
depend on i: 1, hence from (5-37):
<lf(O) e,as
a:rl = o. (5-40)
Then equation (5-39) becomes:
w (bxV f(l) )•c + J(O)(f(l) ) + J (f(l) ) + J(O)(f(l) ) ce - c e,as - ea e,as ee e,as ei e,as
Page 85
c • [ V'ln( p ) + - e,as
m c2 ( e 2kT(O)
e,as
-78-
- l)V'ln(T(O) ) 2 e,as
2C n(O) u(O) ei i,as-i,as
kT(O) c 3 e,as
eE' +(O') +
kT e,as
jf(O) e,as' (5-41)
where appendix A and expression (5-37) have been used to
evaluate the right-hand side; ~i~~s is a diffusion velocity,
see (2-8). The isotropic part of this equation simply reads:
J (f(l) ) = o. ee e,as (5-42)
The function f(l) is assumed to be expanded as in equation e,as (4-58). The general solution of equation (5-42) is:
f(l) e,as (A + BcZ)exp{-
m cZ e }
2kT(O) ' e,as
see chapter IV section 4. In (5-43) A and B are as yet
arbitrary functions of space and time. The choice:
(1) n e,as
= T(l) = 0 e,as '
(5-43)
(5-44)
makes them zero. Then the conclusion is that there is no first
order isotropic correction in a strongly ionized gas:
1<1> = o. e,as (5-45)
From equation (5-41) it appears that f is proportional to ~ e,as
only. The magnetic field makes it necessary to separate the (1)
components of f in the following way: -e,as n(O)
f~~~s = e,as l{f n!11 + f1!1 + ft!t }k, <5- 45 > 1611v.fe k
where: A = bxA. -t - -
(5-47)
The vector A stands for one of the vectors between braces in
( 5-41): Vln(p(O) ) e,as '
Vln(T(O) ) e,as •
eE'
kT(O) e,as
u(O) • The summation -i,as
Page 86
-79-
in (5-46) is over these different possibilities. The general
form of the equation determining f(l) then reads: e,as
w oitf. + lJ (fi) - {-1-+ ce i ee '(l)
2C n(O) ei i,as}f m c3 i
e
1611v3 f(O) ~~~T~e..,......e.,a_s_ b(c),
n(O) e,as (5-48)
where i = 11,1,t; the subscript k has been omitted and oit is
the Kronecker delta. If A = Vln(p(O) ) then b(c) = 1, and so - e,as on, see equation (5-41). Equation (5-48) will be dealt with
further in section 4. The operator 1
J was defined in (4-65). ee
Without the term (<(l)(c))- 1fi the equation is identical to the
equation that has been solved numerically by Spitzer and Harml.
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:
(5-49)
In the limit <0+ ~, the isotropic part of the second order
equation reads:
df(O) e,as
d<2
m c2 + _e __ f(O) V•w(O)
3kT(O) e,as -a,as e,as
eE' ' - --- • ~(c3f(l) )
3m c 2 ac -e as e ,
J(2) (f(O) , f(O) ) + fJ. J (f(l) •c, ll) •c) + J (f(2) ) + ea e,as e,as O ee -e,as - -e,as - ee e,as
f,J(l)(f(l) •c f(O) ) + P,/2)(f(O) f(O) ) 0 ei -e,as -' i,as 0 ei e,as' i,as • (5-50)
where P0 is the operator which when operating on some function
gives the isotropic part of that function. In general (see also
appendix C):
0 __ (2n+l)!! f d n r Q <c >. n 411nlc2n Q c -
(5-51)
c
Page 87
-80-
Equation (5-50) is an equation for the isotropic correction
f(Z) and is of the same type as equation (4-97) for the first e,as
order isotropic correction in the case of a WIG.
When ; 0+ ® the nonisotropic part of the second order equation
reads:
3f(l) c• _:.e,as + <cc>:Vf(l)
"w(O) ( l) m c a (O)
-<cc>•f + -ko- • _:.a,asf - 3; 1 -- -e,as m c
e -- -e,as kT\VJ a, 1 e,as
m + ~e~ f(O) <cc>:Vw(9)
kT(O) e,as -- -a,as e,as
- w c-(bxV f(Z) ) ce- - ·c e,as
e,as
= 3 (0)(f(2) ) + J (f(2) ) + 3 (0)(f(2) ) + 3 (l)(f(O) f(l) ) + ea e,as ee e,as ei e,as ei e,as' i,as
<cc>:fJ.. [/:)(f(l) •c f(O) ) + J (f(l) •c,f(l) •c) + 2 ei -e,as -• i,as ee -e,as - -e,as -
+ J(2)(f{O) f(O) )j ei e,as• i,as (5-52)
Next f(Z) is expanded into irreducible harmonic tensors. Then e,as equation (5-52) can be separated into two equations: one for
f(Z) and one for f(Z) • The collision terms are evaluated with -e,as =e,as the aid of appendix A. The equation for f(Z)
-e,as is of the same type as the one for f(l)
-e,as and reads:
w bxf(Z) + J {f(Z) ) ce- -e,as 1 ee -e,as -
2C n(O)
1_1_ + ei i,as }f(2) '{l) m c3 -e,as
e
2C n(O) u(l) ei i,as-i,as
kT(o) c3
a£< 1 > f(O) + _:.e,as e,as a. l
m aw(O) + e f(O) -a,as ~ e,as a:F1 • kTe as e,as
' The equation for f(Z) takes the following form:
=e,as
2w bxf(Z) ce- =e,as + J ci2 > > 2 ee =e,as -
6C n(O) {-1- + ei i,as}f(2) '(2) m c3 -e,as
e
(5-53)
Page 88
Vf(l) -e,as
-81-
- _eE:_' f{l) + _m_e_ f(O) V'w(O) + mec -e,as kT(O) e,as -a,as
e,as C n(O) u(O) ei i,as-i,as{2_ + 4 .~}f(l) _ f> J (f(l) •c f(l) •c).
me c5 C'+ac -e,as 2 ee -e,as -'-e,as - (5-54)
When T 1+ ~ these equations simplify further since then u(O) =O. -i,as
Equations (5-50),(5-53) and (5-54) can in principle be solved
if the first order contributions fi(l) and f{l} are known. The. - ,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 nonisotropic
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.
Page 89
''
-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)
an(O) e
a:t'o
and (4-69).
aE:< o) e "' O, = a:t'o
The zeroth order equations now read:
(0)
m n (0) {aye + w u (O) xb} + J(-1- + e e aT 0 ce-e - T(l)
In first order of e: an(O) an(l) ~e + e + V•(n(O)Y(O)) = O, cT
1 1T
0 e e
I ( 1 =-l--+
T(l)
( 0) (0) 3<cc> (0) - 2C n u. •J-=- f d3c
ei i -i c5 e '
(5-55)
= o. (5-56)
(5-57)
(5-58)
(5-59) (0)
When T0+ ~, f has become isotropic, which causes several e,as terms in these equations to vanish. The equations then read as
Page 90
-83-
follows:
an<0> aT<0> e,as e,as = O,
a;:l = a;:l
V (O) + en(O) E' + m n(O) w u(l) xb Pe,as e,as- e e,as ce-e,as -
-f(-1- + '<t)
2C n(O) ei i,as)m cf(l) d3c + m c3 e- e,as
e
where: vei = ni~~sce1 /(2mevfe)·
m n(O) v u(O) 312if e e,as ei-i,as'
(5-60)
(5-61)
In contrast to the case of a WIG this equation does not 'close'
when '(l) 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 -r 0+ 00 :
an(O) e,as + V•(n(O) u(l) ) = O,
a:£2 e,as-e,as (5-62)
au< 1> ...:::..e,as +w u(2) xb +f(-1- +
2C n(O) cf( 2) Bv u{l) ei i,as)- eoasd3 + ei-e,as 0
a-r2 ce-e,as - '(1) ( ) c = • m c3 n 312'iT
e e,as (5_63 )
dT(O) 3 (O) k{ e,as + u{l) •VT(O) } + V•(g_(l) + P(O) u(l) J + "t°e,as crr2 -e,as e,as e,as e,as-e,as
+ en(O) u(l) •E' + p(O) V•w(O) e,as-e,as - e,as -a,as
- m c2 e + kT(O) ~(-1-) ]f(O) d3c,
a,as ac '(l) e,as (5-64)
See appendix A for the evaluation of the moments of the
collision integrals. When use is made of the fact that f(O) is e,as
a Maxwellian further simplifications can be obtained.
Page 91
-84-
If the local entropy density is defined as:
s(O) e,as
= -k/f(O) ln(f(O) )d3c, e,as e,as
it is possible to write the energy equation (5-64) in the
following way:
(5-65)
as(O) g _e,as + V•{~ + (u(l) + w(O) )s(O) } =
i (l) •X' + (l) •X -e,as -m Se,as -q
aT 2 TtOJ -e,as -a,as e,as (0) Te,as
m T(O) +~(~
ma T{O) e,as
e,as
m c2 l \.l..._ f~(O) d3c +
'T(O) T(l) e,as e,as
Bm v T(O) e ei (0) (:fof-i as p -
m lfi e,as T 0 i e,as
1) +
(5-66)
where i(l) X' and X are defined, just as in chapter of the -e,as -m -q WIG, as:
i(l) -e,as
:=> m n(O) u(l) e e,as-e,as X'
-111
X "' - Vln(T(O) ), -q e,as (5-67)
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 Ti"+-"' it is clear from
(5-64) that in the case without a magnetic field the correction
f(l) had the general form: -eA
where w is now the new independent velocity variable:
m c2 e
w • 2kT(O)' eA
(5-68)
(5-69)
The functions ~ and B are solutions of the following equations
(see also appendix C and next section):
Page 92
-85-
iA = w312, (5-70a)
.t'B = w3/2{w - 5/2), (5-70b)
where l is a symmetric, negative-definite integral operator.
With the definition of the fluxes !~~) and g~~) and the infor
mation just given the relations between the fluxes and the
forces read:
{5-71)
.. where: D(l) = -D
00
JAZAdw e '
D(l) = -D fBlAdw T e '
v2 D =~
e 3v Ii 0 0
A. ( 1) = - A. 00
JBtBdw e •
0
A. = e
(0) 4 meneA vTe
3v Ii ee
ee
(5-72)
One can clearly see that D(l) and A(l) are positive-definite
and that the Onsager reciprocity relations hold:
m n(0)0 (1) = -2,(l) e eA T vTeAD • (5-73)
The entropy production rate is equal to:
m n(O) ~l~ { e eA 0(l)x2 + T(O) v2 m
eA Te
{5-74)
which is proportional to:
00 X 2 "" X •X oo
- [J~Adw(.....!!!_) + 2fAiBd~ + fBhdwx2]. o v2 o v2 o q
Te Te
(5-75)
This expression is positive-definite if:
00 200 a)
{f~Bdw} - JAlAdwfUBdw < O, (5-76) 0 0 0
which can be proved with the aid of the Schwartz inequality.
In the following section the solution of the equations
(5-70a,b) will be decussed in detail.
Page 93
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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 £ only. It is also clear from the equatidns
that the inclusion of a magnet.le field does not introduce extra
difficulties. In case of a zero magnetic field equation (S-48)
for f = f 11
reads (see appendix D):
w 00
l,t = j'(~5/2 _ ~3/2)f(x)dx +(~5/2 _ -1.,312)ff(x)dx + zw312f(w) 5 3 5 3
0 w
+ ~w(2wF(w)~!) - v(w)f(w) = w312b(w), (S-77)
;; ( w312 where: v(w) = 4V'"'" "ei + )exp(w).
ee '(l)(w)/2 · (S-78)
The function F(w) was defined in (4-108), see also appendix D.
The problem is now reduced to solving euation (5-77), or:
Zf = (~ - v)f = w312b(w), (5-79)
where ~ is the part of 1. coming from the e-e collision term.
The other collisions are present in the function v{w). One can
easily verify that :t and :i are symmetric operators. When i is
differentiated once a pure differential-equation of the Sturm
Liouvil le type is obtained:
-wa a -w JJg = e ~U) , f = aw(ge ) , (5-80a)
a2 -zw a2g] a -2w la ~g = awz[2wF(w)e p - aw[4F(w)e awl· (S-80b)
In the same way an operator$ can be obtained from.,l;:
~ -w a "" i>g = e i;(.l'f)'
a -w f = a;<ge ), (S-8la)
a2 -2w a2g a -2w la aw2 [2wF(w)e p] - aw[(4F(w) + v(w) )e awl +
-w a ( -w) + e aw v(w)e g{w), (5-Slb)
Inspection of this operator shows that the last term of it
vanishes when the plasma is fully ionized, since then v(w)« ew.
Page 94
-87-
For that case, which is referred to as the Spitzer problem, the
operator ~s is defined as followed:
h .. li aw
where ~ • n z2/n is the ionic charge number. i i e
(S-82a)
(S-82b)
The accepted method to solve (S-77) is through an expansion
into a finite number of orthogonal Sonine-{Laguerre-)
polynomialsll, 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$ is
not symmetric in all cases. By means of partial integrations
the following relation is obtained, valid for functions that
are bounded at w=O:
00 df
ffi3)f 2dw • [v(w){f 1(w)0w
2
0
(5-83)
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:
g(O) = 1 s(l) = 5/2 - w 3/2 • 312 • 8(n) = I r(n+5/2)(-w)k
312 k=O (n-k)!k!f(k+5/2) '
fs(n)(w)S(m)(w)w312e-wdw - f(n+5/2) o 3/2 3/2 - n! nm" (5-84)
0
With the expressions for l given in (5-77) it is possible to
calculate the coefficients:
x pq fwpe-w l<wqe-w)dw. 0
(5-85)
This is done in appendix D. The approach is some what different
from that in the litterature: the calculations presented in
Page 95
-88-
appendix D are valid for arbitrary values of p and q, whereas
in the other calculations p and q are restricted to integer
values21 7111. In terms of the operatori>' the problem stated in
(5-79) reads:
ig = e-w ;w(w3 12b(w)). (5-86)
By means of partial integrations one can show that if g 1 and g 2 are solutions of (5-86} with corresponding source terms b 1 and
b 2 and if f 1 and are the related solutions of (5-77) the
following identity holds: .. ff 1~f 2dw = fg
1i{g 2dw, (5-87}
0 0
which directly gives the transport coefficients in (5-72) in
terms of the solutions of (5-86).
The matrix elements for the operator$ are defined as:
6 pq (5-88)
which are also given in appendix o. The calculation of these
coefficients is easier than for A ; they are also valid for pq 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 ii 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:
"" rJi "I' .;;, 1 "' w 0 01 = - 4- + 0 10 = 4l\; + ~ fCw-l)w 2e- Q(w)dw).
0
(5-89)
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
~!tg w312'(l)(w) = O. The parameter 6 is defined as:
n(O) C n(O) TV /2 = __!!!.._ ee eA lnA
ee n(O) 2m v4 Q = (0) L;':;;-aA e Te o naA
where rL is the Landau length.
(5-90)
Page 96
I
-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 $n(w):
N gN(w) = l a $ (w)
n=O n n (5-91)
All these functions $n satisfy the boundary conditions and the
constants an are then fixed by the requirements: 00
ff4:'g - e-w .!__lrw3/2b(w))}$ (w)dw = O· k•O,l, .. .,N. ~ N ~ k ' (5-92)
0
If the functions to be chosen are $n = wn, the equations (5-92)
take the form:
N "' - -wa(3 ] l o a = Je ~ w 1 2b(w) $ (w)dw
0 kn n aw k ' n= o (5-93)
so that the matrix of the equations is not completely
symmetric. It can, however, be made symmetric if the first
equation (k=O) is replaced by:
"" f{.:if~ - w3 12b(w)}e-wdw = O, (5-94) 0
where l and fN are related to ii and gN according to (5-81)
respectively. The function e-w is the solution of the homo-
geneous equation f=O. Therefore:
"" .. 00
-w'# J -w J -wa(-w) fe .,yfNdw = - v(w)e fNdw = - v(w)e aw e gN dw. (5-95) 0 0 0
-wa -w ~ Integrating by parts and using: e aw[v(w)e ] =~l
one obtains:
""1 -w - -r.l"i ~ "'1 ~ e ,tfNdw = ~0 + l a tp .t>ldw o n=Onon
(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=O-equation of (5-93) both
Page 97
-90-
hold reads:
(5-97)
This relation, however, foll9ws directly from the integro
differential 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 operator s •
~ • In terms of this operator the problem then reads:
s 2 ( -w f;) 1J p = f1i F(w)e - 4 , 2
p(w) = - h(w), 3f;
(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 + 00 the right-hand side of (5-98)
becomes zero. On the basis of the general relation for the
diffusion velocity u(l) the first order electrical conductivity -eA is equal to:
6mn v3 e: 212ii 0(1) = K e Te o
e 2lnA (5-99)
where oei is the Lorentz conductivity of a fully ionized
plasma, i.e. taking only electron-ion coll.isions into account.
The constant K is related to the solution of (5-98) as follows:
00 00 w K = .!. - _i fp~6pdw = l+ Jw~e-w fp(w')dw'dw,
z; liio z; o o · (5-100)
which again shows that the conductivity is always positive as
the operator ~s is negative-definite. The exact value of K has
been calculated numerically by Spitzer and HMrml and is equal
to 1.975 if z;=l.
An approximation with polynomials can be made as follows:
N p(w) l n (5-101) "'PN = a w •
n=O n
If N is not too large there is no need for Sonine polynomials.
Page 98
-91-
For N=l a system of two equations for a 0 and a 1 results,
leading to:
K = .!_ + _....;l..;;.5..;..3"'"i; _+--'36.;;..0'-12-"2 __ i; 64i;2 + 244i;l2 + 288
(5-102)
To obtain the same result with the operatori, which has been
done by Landshof2 and Kaneko3, 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:
N n/2 l anw •
n=O (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 HMrm result, see table (5-1). If N=l the
result for K with approximation (5-103) becomes:
13511 - 3211 + ~ + 1;(256 - 71r) 1 412 9 9 K = -+ .,..,...,..,...,..,...,..,...,..,...,..,...,..,...,..,...,..,...,..,...,..,...,..,...,..,...,..,..~ i; (15 + z3 i; + 4i;2)11 - (2 + i;) 2w2
12
N Landshof 2 App.(5-101) App.(5-103)
1 1.9320 1.9498 20
2 . 1.96 , 1
3 1.9616 1.9657 1.9757
(5-104)
table ( 5-1):
values of K
for 1;=1.
Near the origin the solution of (S-98) can be represented by a
Taylor series in powers of w112 which could be an explanation
for the good results obtained with approximation (5-103).
Page 99
-92-
This section is concluded with an examination of the limit of
very small degree of ionization. Equation (5-79) can be written
as follows (see also (4-106)):
.fi 2 ( ) wf( ) 3/2b( ) (" - r;;4.fiew)f(w). 4"6wQwe w =-w w +"' (S-105)
If the degree of ionization is small B 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:
f(w) = l f (w)Sn, n
where:
n=O -w
fo = _ 413 b(w)e
./n Q(w)./;
(5-106)
-w f = _e __ (r;;ew - .L Z}f , nH.
n w2Q(w) ./i n-1
(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(Al) in case of a WIG. Thus the connection with the -e weakly ionized gas theory has been verified.
Page 100
-93-
References
1. L.Spitier and R.H~rm, Phys.Rev. 89(1953)977.
2. R.Landshof, Phys.Rev • .z&.(1949)904, ~(1951)442.
3. S.Kaneko, J.Phys.Soc.Japan ..!2.(1960)1685, .!2.(1962)390.
4. R.S.Devoto, Phys.of Fluids 2_(1966)1230, .!Q.(1967)354,2105.
5. W.L.Nigham, Phys.of Fluids .12(1969)162.
6. C.H.Kruger,M.Mitchner and U.Daybelge, AIAA J. 2.(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 .!Q.(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.
Page 101
-94-
VI NUMJ:'RICAL RESULTS
In this chapter the results of chapters IV and V are
applied to several practical situations. The shape of the
isotropic 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 kij' 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.
Page 102
-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 Jk 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:
(6-1)
The solution of the homogeneous equation is then added after
the constants c1 and C2 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 Gk; see (4-107).
The reference cross section Q0 has been chosen io-20 m2, so
that the dimensionless functions Q(w) and hence the isotropic
functions are uniquely determined.
Characteristic for all isotropic correction functions is the
rather large peak near w=O 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 kij 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
/
Page 103
-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
Ramsauer 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, l.J
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
~he 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 Haxwell inter11ction
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:
Page 104
-97-
seems to be less suited for a description of the electron-atom
interaction than the hard spheres model.
j+
0.40 -1.26 0.81 -3.43 3.96 -9.29
-0.87 2.35 -1.75 8.02 -7.63 21.9
-0.87 2.35 -1.75 8.02 -7.63 21.9
-6.41 15.2 -12.8 63.9 -5o.9 176
-0.10 0.30 -0.21 0.93 -o.96 2.52
0.93 -2.50 1.85 -8.49 8.11 -23.1
Table (6-la): kij constants for hard spheres model (HSM).
.078 -2.74 0.14 -1.25 6.99 -3.59
-0.30 7.78 -0.77 6.15 -20.2 17.9
-.046 1.14 -0.12 0.95 -2.98 2. 77
-0.57 12.4 -1.63 12.0 -32.6 34.7
-.0026 .075 -.0058 .046 -0.19 0.13
.046 -1.19 0.12 -0.95 3.10 -2.75
Table (6-lb): kij constants for neon at Te• SOOOK.
-4.59 -101 -45.5 16.1 208 102
21.9 113 56.0 -56.2 -226 -134
-4.30 -54.4 -25.6 12.5 110 57.0
14.2 148 69.9 -39.9 -301 -158
-3.16 -7.16 -5.05 7.16 14.3 11.1
7.16 23.2 13.7 -16.6 -44.3 -30.5
Table (6-lc):kij constants for argon, Te• SOOOK, data Milloy2
Page 105
-98-
j+
H 3.32 -78.1 -35.4 7.48 160 87.6
11.2 112 64.9 -43.7 -215 -147
-0.79 -52.Q -27.3 10.9 103 63.5
1.00 133 67.5 -26.1 -264 -159
-2.30 1.01 -2.15 4.39 -4.69 2.50
4. 39 11.9 11.0 -10.9 -18.7 -21.l
Table (6-ld): kij constants for argon, Te= 5000K,
data Frost and Phelps3,
k12 = -1.26 k14 -3.43 k23 = -1.75
k52 = 0.30 K3 = -13.6 K7 = 5.02
k22 = 2.35 ksi. - o.93 k21 = -0.87
Kl = 149 KS = -2.61 k25 = -7 .63
K2 - 31.1 KG = 0.38 y 15 = -62.5
k2lt = 8.02 ks1 = -0.10 y 25 = -7.93
Kif = 14.1 kl I - 0.40 Y35 = 7.63
Table (6-2): Some coefficients for the HSM appearing in
equations (4-132)-(4-135).
Page 106
-99-
1 a
0 ../('
:i:
~ lH -1 HSM
k:1
-2 0 1 2 3 4 5 6 1 B
w
4 b
3
2 ,.If'
1 :i:
~ 0 lH
-1 HSM k: 2
-2 0 l 2 3 4 5 6 7 8
w
8 c
4 ,.If'
:i:
'i' lH 0
HSM k::3
-4 0 1 2 3 4 5 6 1 8
w
Fig.(6-la,b,c) Isotropic correction functions for the HSM.
Page 107
-100-
20 d
16 12
8 ..N";;: 4
'i" 0 4-l
-4 HSM -8 k=4
-12 0 1 2 3 4 5 6 7 8
w
.4 e
,3
.2 ..N";;: . t ~ 4-l 0
HSM - . 1 k:5
- .2 0 2. 3 4 5 6 7 8
"' 2
1
0
'i" -1 4-l -2 HSM
-3 k::6
-4 0 1 2 3 4 5 6 7 8
w
Fig.(6-ld,e,f) Isotropic correction functions for the HSM.
Page 108
-101-
a
0
'i: -1
4-< NEON
-2 le='5000K k:l
-3 0 2 3 4 5 6 7
w
8 b
4
'i: 4-< 0
NEON Te-=-5000K
k=2 -4
0 2 3 4 5 6 7 w
7 c
5
3
~ k;. 3 4-<
-1 NEON Te=5000K
-3 0 1 2 3 4 5 6 7
w
Fig.(6-2a,b,c) Isotropic correction functions for neon 1,
at T "' SOOOK. e
Page 109
-102-
12 d
8
4
:; 0 ~ .... NEON
-4 Te:::SOOOK k= 4
-8 0 1 2 3 4 5 6 7
w
. 1
../(" ::: 0 'i' ~ .... NEON
Te =5000K k=5
- . 1 0 1 2 3 4 5 6 7
w
1
0 ../("
::: 'i' ~
\H -1 NEON Te= 5000K
k=6 -2
0 1 2 3 4 5 6 7 w
Fig. ( 6-2d, e, f) Isotropic correction functions for neonl,
at T = SOOOK. e
Page 110
-103-
8 a
4
0 ..r:i: -4 ~ ti.-! -8 ARGON
-12 Te= 5000K k=l
-16 0 1 2 3 4 5 6 1
w
16 b
12
8 ..l'
:i: 4 ~ ti.-! 0 ARGON
-4 Te" 5000K k:2
-8 0 1 2 3 4 5 6 1
w
26 c
20
..l" 12 :i: ~ 4 ti.-!
-4 ARGON Te" SOOOK
-12 k:3
0 1 2 3 4 5 G 7 w
Fig.(6-3a,b,c) Isotropic correction functions for argon,
at T = SOOOK. e
Page 111
-104-
20 d
16
12 ..r':;: 8 ") 4 'H
0 ARGON -4 Te"SOOOK
-8 k~4
0 1 2 3 4 5 6 ? w
1
0 ..r':;:
~ 'H -1 ARGON
Te=SOOOK k:5
-2 0 1 2 3 4 5 6 7
w
4 f
3
2
..r':;: 1 ")
'H 0 ARGON
-1 Te=SOOOK
-2 k:6
0 2 3 4 5 6 7 w
Fig.(6-3d,e, f) Isotropic correction functions for argon,
at • SOOOK.
Page 112
? ~----~....-....... --.......-~
5
3
1
·1
·3
-5 .. ooo 5000 6000 7000
TUP. (KELVIN)
.2 ~----~....-....... --.......-......
.1
0
-.1
-.2
•• 3
.,4 '---'----'~_,__,_ __ ....____. 4000 5000 6000 7000
'\' E1'r • ( KEL Vlllll
60 ..--.............. ~...-_,..--..--.
50
40
30
20
10
0 4000 5000 6000 7000
TEnP. CKl!L VIN)
-105-
10 .-----...--.......-....... __,
8
6
4 z r-;;:.__--_j 0
·2 4000 5000 6000 '1000
TEMP. (KELV1Nl
.s ~~--~....-....... --.......-~
.4
,3
• 2.
.1
0
··1 4000 sooo 6000 7000
TEftP. <KELVIN)
1 . 8 ....-....-....... --..---....--.-.,,,.,., 1-4
1 .6 • 2.
•. 2 i:----=.:..... __ -.J -.6 ·1
·1·4 L--'-__Jl.....--1....--1. __ ..i....;::::.i
4000 5000 6000 7000 TEl'll'. om.VIN)
Fig.(6-4) Several coefficients appearing in the transport
coefficients for neonl; see equations (4-132)-(4-135)
Page 113
~io6-
••
. 5
.4
.,
.2
b: E•300V/m .1
0 0 1 2 3 5 6 '1
w
.&
.s
...
·' .2
a:E=1SOV/m .1
0 0 2 3 5 & 7
w
Fig.(6-Sa,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 1018m- 3 ,T = T = 5000K. a e e a
M: zeroth order Maxwellian without isotropic correction.
Page 114
-107-
-6
.s
... . . 3 ' '\ .2 "· " . ~ .1 "· --=.
~.
0 0 2 3 4 s 6 1 8
w
Fig~(6-6) Effect of a temperature difference on the isotropic
part of the electron distribution function in case of
argon2: p • a
T • e
1 atm., n • 3.4 1017m-3, e
4500K, Ta• lOOOK.
M: zeroth order Maxwell~an without isotropic correction.
Page 115
-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 s 0 ,R0 and L 0 which are functions of
the electron temperature except for the case of the USM. 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:
(6-2)
so that:
-(m+l)/2 T Q(w) = q=w , q = ------~
•u "lll T (2v )m/2 m Te
1 'T = -----
n r-f v~ Q0 a Le
(6-3)
The coefficients mentioned above are then easily calculated
giving the following results:
(6-4)
The hard spheres (!ISM) corresponds to m=-1 with q- 1=1.
The following coefficients can be calculated exactly for the
interaction model (6-2):
s ee
L ee
SA. R -2 (A. m/2 ,m/2)
ee = qm m/2,m/2+1- 2 '
2SA. -2(;i. _ SA. + m/2,m/2)
qm m/2+1,m/2+1 m/2+1,m/2 4 ' (6-S)
where the coefficients A. are defined in (D-41) of appendix p,q
D3. A tedious but straightforward calculation gives for the
USM:
s - [_!2_ + .!i - lln(i+l2> ]li = -0.2216 ee 3012 lS 4
Page 116
-109-
R - [~ - .!.!. - .!1..1n( i+l2)] ;;r = o. 6436 ee 6012 30 8
[1093 34 95 ~ ] I Lee = -- - IT+ 161n(l+v2) >'!! = -1.8775 12012
(6-6)
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 Ramsauer 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:
w -x -x J _e_ l{-e-}dx, o Q(x)IX Q(x)IX
(6-7)
which is related to one of the coefficients in (4-121), namely
s ee SE( 00). One can then see that the main contribution to the
integral See comes from the Ramsauer minimum. It is clear that
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 B, defined as:
n r 2
B = v 1:12 = ~~....!:. ee na ~11 q0 (6-8)
where rL 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 B is of the order £, up to second order
reads:
BS .+ ~ }. (6-9) ei .;-; ee
Page 117
-110-
From this expression one can infer that when S 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. 4 , which is defined as follows:
1 1 1 -- ·=-- +---0add • o(l) YE 0 ei '
(6-10)
where o(l}= o0s0 is a result of electron-atom collisions only
and o of electron-ion collisions only: ei
0 ei := (6-11)
and YE is the well-known Spitzer factor: YE = 0.582, so that
yEoei is the electrical conductivity of a fully ionized
plasma5, Mixture rules proposed by Frost&, 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: -w
J w5 12e dw
ao o {w313q(w)+0.952S}
(6-12)
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-
Page 118
-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:
(6-13)
The function w112 is not permitted because it results in
infinitly large matrix elements o • In table (6-3) values of - pq
-(A,:A) 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 Polynomials Functions
functions in (6-13)
2 1.3393 1.3393
3 1.3842 1.4069
4 1.4128, 1.4370
5 1.4264 1.4401
6 1.4325 1.4402
7 1.4354 1.4402
8 1.4370 1.4402
Table (6-3), values of -(A,tA) for the HSM with 8=1.
When 8 + 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 o0s0 for low values of S and to the
Spitzer conductivity yEoei for the higher values. The relation
between these normalizations is the following:
(6-14)
Page 119
-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 BSM
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:
+ iqL - k R ) } Ii ee ---r ee ~
where K4 = k 26 - kTk25 - \k24 •
-1)-il(L -kR.)+ ei T ei
(6-15)
For large a 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 a the normalization is done with
respect to (L 0 - kTR0 )A0
, i.e the first order contribution.
When Ta/Te = 0.9 and Te= 5000K, the thermal heat conductivity
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 K4
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.
Page 120
-113-
0.1
no1...._~ .................. _.__._ ......... ..__..._ ............... OJ)l 0.1 10
cV
Fig.{6-7) Plot of the data for the electron-argon cross section
for momentum transfer a:s obtained by Milloy2{--
and by Frost and Phelps3(- - - -).
l.iJ (J)
Fig.(6-8)
10
0
-10 -20 -30 -40 -50 -60 -70 -60
0 .s 1. 5 2
Plot of the function SE(w), see (6-7) for the cross
section data of Milloy(M) and of Frost and Phelps(FP)
at T = 5000K. e
Page 121
-114-
t . I
. 9
....sL .8
u..s • HSM
. 7
cE "'0.01 .e nakT•Qo
.s 10·• 10" ·• 10
_, 10• 10
B Fig.(6-9a) Electrical conductivity normalized to the zeroth
order value a S for the HSM. 0 0
.9
.8
.'1
...!!... •• O'Sp .s
.4
,3
.2
. 1 HSM
0 0 10
1 I. ul 104
10 10 B
Fig.(6-9b) Electrical conductivity normalized to the value of
the fully ionized plasma for the RSM.
B = 2\v ~ , AR: Addition mixture rule, FR: Frost mixture rule. ee
Page 122
-115-
1 . t
.9 ' . ' ' .e ...!L 11 • .s • ,7 Te=SOOOK
. s _!.L = 0.01 n.akTe'lo
.5
.4 10·4 10·1 ·1 -1
10° 10 10 6
Fig.(6-lOa) Electrical conductivity normalized to the zeroth
order value for neon at T = 5000K.
fl
O''~
t
,9
·8
,7
.s
.s
•• . 3
.2
. t
0
e
NEON fe•SOOOK
10° 104
B Fig.(6-lOb) Electrical conductivity normalized to the value of
the fully ionized plasma for neon at Te• 5000K.
~ = 2\v ~ , AR: Addition mixture rule, FR: Frost mixture rule. ee
Page 123
-q6-
1 .1
.9 ..... a: .. .a .M
~ I
• HSM _, - .7 0
.< .6 Ta r -o.9
e .s
.4 to-• -3 -2 -I 0
10 10 10 10 B
Fig.(6-lla) Thermal heat conductivity normalized to the zeroth
order value for the RSM •
. 9
. 8
,7
.s
.!.. .5
xs,. ,4
.3
.2
. t
0 D
10
.&
FR" ~
" SIG /,
HSM
B Fig.(6-llb) Thermal heat conductivity normalized to the value
of the fully ionized plasma for the RSM.
2~v ~ , FR: Frost mixture rule. ee
Page 124
-117-
1.2
t . t
1
..... .9 a: :it' .e
~ I
..r - .?
.<.o .6
NEON .5
i: .4 ta =0.9
t
.3 ·• -a ·2 10·1 0 10 10 10 10
Fig.{6-12a) Thermal heat conductivity normalized to the zeroth
order value for neon at T • SOOOK. e
.9
.e
.7
.s
_1_ .s
Asp .4
.3
.2 NEON
.1 T~ 5.000K
0 l o0 Io' 10
2 10
3 1 o4
Fig.(6-12b) Thermal heat conductivity normalized to the value
s of the fully ionized plasma for neon at Te~ SOOOK.
2~v ~ , FR: Frost mixture rule. ee
Page 125
I
-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 8 outside the scope of the present theory of the WIG.
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 Postma 7 and the other one by StefanovB. 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
Stefanov's cross 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 Boltzmann
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
a1.10. 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 ~ixture 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 Ramsauer minimum of argon
an<l thus reinforces the influence of the latter.
Page 126
-119-
f~ ·- « i~ /
.. .:. .. 2 ' ..... I Fig.(6-13) electron-cesium momentum-.! . .. \
I.: transfer cross section
-:: data as obtained by .. • a .Jr...J v1foc .. ~ 7 • c 10• "'"''
Postma 7(P) and by Stefanov8(s).
101 ia
,,,-+ /~
;: "' • / .. 11 ~ 10 ..-:: ... .. .6 ... ., :c
°' ' ' "" 0 :c "' ~ !S
" .,; 10·1 0 ·1
"' ,. 10
C1 .. ... ... 0.1 torr Cs
Postma Stefanov
10·• 10·• 1300 HOO 1500 1600 1'100 !BOO 1300 1+110 1&00 11100 !700 1800
TEllP. OlcLYINl TEMP. !KELVIN)
Id 10' /
•-" /
• / .. 10• "' 10°
,,, w ~ / .... ... .... ~ t·
' 0 0
"' "' c !;
.; ·• .; ·• "'
10 z 10 .. 8 .. 1 Iott Cs Stllfanov
·• -· 10 10 1300 1400 1500 1600 1100 1800 1300 HOO 1500 1600 1700 1800
TEnP .. tKELYINJ TEnP, CKELVINl
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 Postma 7 and of StefanovB.
argon pressure is 1 atm.
Page 127
-120-
References
l. A.G.Robertson, J.Phys.B 2_(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)Al538.
4. S.C.Lin,E.L.Resler and A.Kantrowitz,
J.Appl.Phys. 26(1955)95.
5. L.Spitzer and R.Harm, Phys.Rev. 89(1953)997.
6. L.S.Frost, J.Appl.Phys. 32(1961)2029.
7. A.J.Postma, Physica 43(1969)465.
8. B.Stefanov, Phys.Rev. A22(1980)427.
9. L.P.Harris, J.Appl.Phys. 34(1963)2958.
10. C.H.Kruger,M.Mitchner and U.Daybelge, AIAA J, 2_(1968)1712.
\
Page 128
-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. As
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 Boltzmann 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
Page 129
-122-
for energy relaxation between electrons 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 isotropie 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.
Page 130
-123-
The calculation of the coefficients needed for the Galerkin
method happens to be easier in this formulation. The 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 case 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 obtai~ed 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.
Page 131
-124-
APPENDIX A
Expansion of the electron-heavy particle collision integrals
A-1 Electron-atom collisions
The Boltzmann collision term describing elasic collisions
between particles of species s and t in a dilute gas is usually
given in the following form:
J (f ,f) = Jd3gJbdbd~g{f (v')f (v') - f (v)f (v )}, st •s t s - t -t s - t -t
(A-1)
where b denotes the impact parameter and ~ is the relative
velocity: ~ = ~t- v. Primes indicate post collision variables,
which are defined by the relations:
v' m JI. t-
y - m +m • s t
v' -t
m !l sv + ---t m +m
s t !/, .. g'- g. (A-2)
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:
Jl.2 + 2~·~ = o. (A-3)
It is now possible to show the integrations to be performed
explicitly if the differential cross section o(g,x) is defined
by:
bdbd~ (A-4)
1where x is the scattering angle in the centre of mass system.
Figure A-1 shows an encounter in that system. With the help of
relation (A-3) the integration over x and ~ is written as an
integration over the complete Jl-space. Expression (A-1) then
reads as follows:
J (f ,f ) = 2Jd3gJd3JlI(g,!l)o(!l.2 + 2g•Jl.)x st s t - -
m JI. m !l
x{fs(v - m !~ )ft(y + & + m ~) - fs(y)ft(y + g)} s t s t
(A-S)
Page 132
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b I abl where: I(g,R.) = a(g,x) = sinx ax • (A-6)
The o-Dirac function takes care of relation (A-3).
The collision integral J describing the collisions between ea
electrons and atoms will be expanded in powers of the small
parameter e::
E: • (m /m )!2. e a (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:
(A-S)
J(O) = /d3v f {v )/d>Na(v,x)[f (u) - fe(y) ], ea a a -a e - (A-Sa)
J(l) = /d3v v f (v )•/d0a(v,x)2(nn - I)•V f (u) + ea a-a a -a -- = v e -
- /d3v v f (v )•/dOV (va(v,x>){f (u) - f (v)}. a-a a -a v e - e - (A-Sb)
J(Z) ~/d3v v v f (v ):/dO{V V (va(v,x))(f {u) - fe(y)) + ea a-a-a a -a v v e -
Page 133
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4V (va(v,x))(I-nn)•\7 f (u) + 4vo(v,x)(=I-.!!!!H_!-_n_!!):Vv\7vfe(~)} +. v = -- v e -
2m + .....!.Jd3v f (v )\7 •Jdn(I - nn)•vva(v,x)f (u).
m a a -a v = - - e -(A-8c)
a
In these expressions ~ is the electron velocity after a
collision with an atom of infinite mass, so that u=v. The unit
bisector of the angle between~ and y is ~· see figure A-2.
tJhat is left from the
integration over i-space
is an integration over
dn = sinxdxd<j>.
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 hydrodynamical 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 a case the third order contribution vanishes also:
J .. /O) + e:z/2) + t}(e:'+) ea ea ea ' (A-10)
(A-lOa)
kT / 2) =~Jdn{a v(v,x)(f (u)-f (v)} + V v(v,x)•4(I-nn)•V f (u) ea ~m v e - e - v = -- v e -a
m + v(v,x)4(I-nn):V V f (u)} + _!!v •Jd'2v(v,x)2(I-nn}•vf (u). •- vve- m v =- -e-
a (A-lOb)
where: v(v,x) = n vo(v,x)· a
The collision operator permutes with the rotation operator in
velocity space. Therefore the spherical harmonics or equi
valently the harmonic tensors of appendix C are eigenfunctions.
If fe is expanded into harmonic tensors:
Page 134
-127-
00
fe(y) • l f (v)•<c°>, n•O n-e n -
(A-11)
and if this expansion is substituted into (A-lOa) the n-th term
reads:
f (v)•JdQv(v,x)(<u°> - <vn> ]. n-e n - -
(A-12)
Consider now the right-hand factor of this n-fold dot product.
After having performed the integration over dQ = sinxdxd~ where
x is the angle between u and y, the result still is a harmonic
tensor of rank n, therefore:
JdOv(v,x)<u°> = <v°>JdQ g <x.~)v(v,x), - n
(A-13)
where gn is as yet undetermined. By taking the n-fold inner
product of this expression with <vn> one can show (see app. C)
that g (x.~) = P (cosx) so that: n n
J(O)( f(v)•<v°>) • f(v)•<v°>fdGv(v,x){P (cosx) -1}. ea n- n - n- n - n
In general with expansion (A-11) for some function f : e
"' J(O)(f ) = - l ,-(ll)(v) f (v)•<v°>,
ea e n•l n-e n -
with: -....;;...,.....- = Jdl'lv(v,x)[l - P (cosx)J. \1) n
(A-14)
(A-15)
Note that the term n•O is absent: J(O) is zero ea
isotropic function. In appendix B it is proved
for an arbitrary
that /O) is ea
symmetric and negative-definite.
derived.
Also an H-theorem will be
The expression for J(Z) becomes also very simple if f is ea e isotropic:
/2)(f ) =me a [~(l + .L)f ]. ea e m vLaV i: m v av e a (1) e
If f is a Maxwellian too, this expression becomes: e
m T 3 /2)(f ) - err- a,1 a [..:!._ f (v)] ea eM - m' T'VLaV 1: eM • a e (1)
(A-16)
(A-16a)
which shows that this term vanishes in thermal equilibrium.
Page 135
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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:
1 ~ V }f (v)f (v')d 3v• m v' a - 13 -
where:
811e:2m o a
13
and C is a constant defined by: a6
(A-17)
(A-18)
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:
(A-19)
/1) ei
c . - ....!::!.fv'f (v')d 3v'•V •[(V V)•V f]
m - i - v v= v e ' (A-20) e
ceini 2v c i / 2 ) = -- v +-=- f ) + _e_fv'v'f (v')d3v•:v •[(V V V)•V f ] ei mi v v3 e 2me - - i - v v v= v e
where the tensor ¥ is defined as follows:
v 2~ - vv ¥ :=---
v3 v v v. v v
Some properties of ~, which are easily verified are:
~·~ = o,
2v V •V
v = - --,
v•V V v=
v3
-v ='
(A-21)
(A-22)
(A-23a)
(A-23b)
(A-23c)
Page 136
-129-
v-(V V V) "' -2V V , - v v= v=
(A-23d)
(A-23e)
v211 = L(v2 .!..) + v3v •(V •V ). v av av v = v (A-23f)
With these properties it can be shown that:
(0) ~ ceini J . (f ) "' - 2 n(n+l)-- f (v) •<v~, ei e n=l m v3 n-e n -
(A-24)
e
and if f is isotropic:
( 1) cei 2! of J (f) = - -fv'f (v')d3v• • --. ei me - i - v4 av
(A-25)
2C n C / 2)(f) =~~- eifv'v'f (v')d3v':<v2>[.2_.+!.....L1of + ei 2 av m. - - i - - 6 5 av.rav miv i v v
+ 2Ceifv'2f.(v')d3v• l a (1 of) 3m 1 - v2 av v av • (A-26)
e
The second term on the right-hand side of (A-26) clearly
vanishes if fi is also isotropic. If the electron and ion
distribution functions are Maxwellians with hydrodynamical
velocity equal to ~a and with different temperatures one finds
that the first contribution to Jei is given by (A-26) and
reads:
i~>(f ) me 2Ceini Ti = ---(- -l)f (c) (£ = v - w ) • (A-27) ei M mi ckT T M ' -a e e
See also equation (A-16a) which is 'of the same type.
Page 137
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A-3 Moments of the electron-heavy particle collision integral
All collision integrals considered are elastic and thus
preserve the number of particles:
JJ (f ,f )d3v = O, st s t
In general the moment with some function ~ of y reads:
JJ (f ,f )~(v)d3v, st s t -
(A-28)
(A-29)
If J is the Boltzmann collision integral of (A-5) and the st
following transformations:
JI. +-JI. - _, a .. s + .£, m JI. t-
v .. v - m-· 0
(A-30)
are applied in this order one obtains for the moment (A-29):
m JI.
x[¢(y - mt-) - ¢(y)]fs(y)ft(y + s>. 0
(A-31)
which indeed is zero if ¢•1. If ¢-ms! equation (A-31) becomes:
2m mt Jd3vmsyJst = - ~Jd3vd3gd3Jl._£o(Jl.2 + 2g•.£)I(g,Jl.)fs(y)ft(y + g)
0
(A-32)
Firstly the integration over !/.-space is performed in spherical
co-ordinates with ~ as polar axis:
(A-33)
where T(l) is defined as in (A-15). Thus:
ms mt 3 B Jd3vm vJ t "'--fd3vd g ( ) f (v)ft(v + g).
s- s m0 ntT(l) g s - -(A-34)
If s=e,t=a this term can be expanded in powers of e. It is then
convenient to perform an integration over !'= y + a instead of
one over &· Th~ result of the expansion for this case is:
Page 138
-131-
- kT V ( \ ))]f (v)d3v +(J(ei+). a v T(l) v e -
Next '(v) = \m v2 is inserted into (A-31): - s
x f (v)f (v + g). s - t - -
With the aid of (A-33) and s similar integral: 2
Jd3 JI. J1.2o( .11.2 + 2,g•.11.)I(g JI.) = _ _,g...._ __ - t UtT(l)(g) >
the moment integral with \m v2 can be written as follows: s
mm m g 2 f (v)f (v + ~) 2 3 st33(t )s-t-., J\m v J d v = --! d vd g -- + a •v •
s st m0
m0
" - ntT(l)(g)
(A-35)
(A-36)
(A-37)
(A-38)
If again s=e and t=a the following expansion in powers of e is
obtained by means of Taylor expansions of the integrand:
m 3kT - m v2 J\m v2J d3v = e~f[ a e + kT v·!...(-1-) ]f (v)d3v + {?( ei+)
s ea m T(l)(v) a av t(l) e -a (A-39)
Finally the tesults are given of the expansions of the moments
of the electron-ion collision integrals:
v fm vJ .d3v = - 2C n J=-f (v)d3v +
e- ei ei i v3 e -
3<v2> 2m v - e2C fv'f (v')d3v••f---f (v)d3v - e~ iniJ=-f (v)d3v +
ei - i - vS e - mi e v3 e -
(A-40)
v J\m v2J d3v • -e2C .Jv'f (v')d3v'•J=-f (v)d3v +
e ei ei - i - v3 e -
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Some caution must be taken with integrals like:
<v2> J-=--f(v)d 3v.
v5 -(A-42)
At first glance this integral would be zero if f is isotropic.
But this is only true if f(O)=O. The result of the integration
must be an isotropic tensor of rank two (see also app.C):
J--f(v)d3v = -31fv 'I (_!.)f(v)d 3v = A(v)I.
v v v =
Contraction on both sides gives:
3A = .!.Ja (.!.)f(v)d 3v = - 4nfo(v)f(v)d3v - 4n3f(O).
3 v v 3 -
<v2> f---f(v)d3v = - 4nf(O) I.
vS , 9 = so that:
A generalization of (A-43) is:
fvn(.!.)f(v)d3v = A (v) I, v v n n-
(A-43)
(A-44)
(A-45)
(A-46)
wliere I is an isotropic tensor of rank n. For odd n there is n-no such tensor so that these integrals are zero. This follows
also from the fact that the integrand is an odd function of !•
If n is even An can be determined by a contraction over all
indices and a subsequent calculation of the integral as
follows:
f n/2 (1) 3 f ( n/2-1 ) 3 Av v f(v)d v = - 4TI Av o(y) f(v)d v. (A-47)
If f can be differentiated a sufficient number of times this
integral can be calculated.
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APPENDIX B
Some H-theorems and properties of collision integrals
B-1 The zeroth order electron-atom collision operator
The zeroth order electron atom collision operator J(O) has ea
been derived in appendix A. In this section two properties and
an H-theorem will be derived. With the aid of the o-Dirac
function the collision integral /O) can be written as follows: ea
J~~)(f) = naJd3J1.o(J1. 2 + 2~·!)I(v,J1.){f(! - ~) - f(!)}. (B-1)
By multiplication with some function g and integration over the
entire velocity space the following Inner product is obtained:
(J~~)(f),g) = naJd3vd3J1.o(J1.2 + 2!·~)I(v,.11.){f(!-!) - f(!)}g(!)·
(B-2) If the velocity transformations ! + v - J1. and J1. + -.11. are
performed in this order, (B-2) is found to be equal to:
(iO)(f),g) = ~n Jd3vd3to(t2 + 2v•J1.)I(v,.J1.)x ea a - -
(B-3)
so that J(O) is a symmetric operator. From this expression one ea finds immediately that J(O) is negative-definite:
ea
(B-4)
Finally an H-theorem can easily be proved as follows. Suppose
that the following equation holds for f:
(B-5)
The quantity H is now defined as:
(B-6)
If equation (B-5) is multiplied by: l+ ln(f),and integrated
over v-space, the following inequality can be obtained:
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n f(v) = 2afd3vd3to(£2 + 2y•!)I(v,£)ln(f(y=!)){f(y-!) - f(y)} < o,
(B-7)
because f is assumed to be positive everywhere. At the same
time H is bounded from below, so that when t. + 00 the integral
in (B-7) is equal to zero and f has become an isotropic
function, i.e. depending only on l!I•
B-2 The zeroth order electron-ion collision integral
The above derivations suggest that the properties of J(O)
also hold for J~~)· They will be proven below. One has: ea
( 0) nicei nicei (g,Jei (f)) = --{gV -(V•V f)d3v = - --JV g•Y•'V fd3v,
me v = v me v - v (B-B)
which is symmetric because X is symmetric. If g=f the
integrand in the right-hand side of (B-8) takes the form:
a2v2 ..: (!•y)2
!!·~·!! = -----v3
;. o, (B-9)
which shows that J(O) is a non-positive operator. Finally an Rei
theorem is derived. Again R is the quantity defined in (B-6).
If in (B-8) g = I+ ln(f) and f obeys:
1f = /O)(f) at ei •
it is easily demonstrated that:
(B-10)
(B-11)
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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 9i is defined by:
(0) (0) ( mi 3/2 f = +i(y)ni as (0) ) exp{-i,as ' 2nkT
m lv-w(O) 12 i - -a,as }
2kT(O) a,as a,as
+1 (y)fiM.
(B-12)
With the results from the moment equations it can be shown that
ClfiM - = O. (B-13) <l-r1
Equation (4-20) is now multiplied by +i(y) and integrated:
a+z f~~fi~3v = 2fd3vd3R.d3go(£2 + 2g•,&.)I(g,Jl.)f~~~s(y+g)fm(x)x
m JI. x{+1<! - ma-) - +1(y)}+i(y), (B-14)
0
where m0
=ma+ m1• Application of the following transforma-
m JI. a-t ions in the given order: ! + -~. ! + y - ;-- , g + s + ! is 0
equivalent to interchanging direct and inverse collisions. With
this transformation equation (B-14) can be written as follows:
aHi (0) aTi = -2fd3vd3R.d3go(Jl.2+2g•£)I(g,Jl.)fiM(x)fa,as<y+g)x
m JI. 2 x{+i(y)-+i(y- ma-)} ~ O,
0
(B-15)
,where: Hi= ffiM+fd 3v. (B-16)
The conclusion is then that +i=l when -r 1+ w, for Hi is non-
negative and decreases with time, i.e.:
I (0)12 (0) mi 3/2 mi y-~aA
niA ( (0)) exp{- (O) }. 2nkTaA 2kTaA
(B-17)
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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 T0+ 00 they read:
of(O) a,as
a:rl J (f(O) f(l) ) + J (f(l) f(O) ) +
aa a,as' a,as aa a,as' a,as J (f(O) f(O) ) ai a,as' i,as
(B-18)
J (f~O) f~O) ) + J. (f(O) f(O) ) • ii 1,as' 1,as 1a l,as' a,as (B-19)
Again f(O) is a Maxwellian. If equation (B-18) is multiplied a,as
by (l+ln(f(O) >)and integrated over the entire velocity space a,as the .following equation is obtained:
where: H(O) a,as
ff ln(f(O) )d3v. a,as a,as
(B-20)
(B-21)
The terms containing J vanish because (l+ln(f(O) )) consists aa a,as of mere collision invariants. Equation (B-19) is multiplied by
(l+ln(fi(O) )) and integrated too: ,as aH(O)
i,as a;: l
where:
+ l(l+ln(f~O) ))J. (f(O) ,f(O) )d3v, 1,as 1a i,as a,as
= Jf(O) ln(f(O) )d3v, i,as i,as
Then the following inequality can be provedl:
(B-22)
(B-23)
(B-24)
(0) from which the conclusion can be drawn that when T1+ 00 , f
1 ,as relaxes to a Maxwellian with a temperature and a hydrodynamic
velocity equal to those of the atoms as in the case of a WIG.
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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:
(-l)nv2n+l n 1 <y°> := (2n-l)!! vv(v)· (C-1)
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 <v°> can be seen as the irreducible part of the n
tensor v := YYY···Y (n vectors). The first few harmonic
tensors written in index notation are:
<vl>. = vi' - 1
(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:
4nn!v2n Jnh(v)~<yn><y~dnv - 0nm (2n+l)!!<nh(v)>,
where n~ is an arbitrary tensor of rank n and <n~> is the
irreducible part of n~· The following expansion is very
useful213 :
(C-3)
n = y n(n-1) Z[I n-2] + n(n-l)(n-2)(n-3) 4 [I 2 n-4] 2(2n-l)v .Y 8(2n-1)(2n-3) v = Y - •••
(C-4)
where the square brackets denote the symmetric part, obtainable
by adding all the permutations and deviding the result by nl.
The inner product of y and <yn> will again be an irreducible
tensor but now of rank n-1, and will thus be proportional to n-1
<v >. This tensor also has an expansion as in (C-4).
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The inner product !•<yl\ is thus equal to some factor times n-1 <v >, which will appear in the right-hand side of (C-4) after
h:ving performed the inner product with!• The tensor y•[!yn-2]
n-1 -possesses 2(n-l)! permutations equal toy , therefore:
n(n-1) 2(n-1)! v2vn-l nv2 n-1 2(2n-l) n! + • • • "' 2n-l <y >
(C-5)
This result can easily be generalized to:
k n n(n-1) ••••• (n-k+l) 2k n-k <y >k<y > • (2n-1)(2n-3) ••••• (2n-2k+l) v <y >, n)k, (C-6)
With the aid of the definition (C-1) it can be shown that:
n 2n+l n n+l V <v > = ~ v<v > - <y >) , v - v2 - -
(C-7)
From which immediately follows with (C-5):
" •<vn> = (2n+l)n n-1 •v - 2n-l <y > (C-8)
If again h(v) is an irreducible tensor the following relation n-
h olds:
v<vl\• h = vn+l. h. -- nn- nn-
(C-9)
n+l With the expansion (C-4) for <y > this equation becomes:
v<vn>• h = <vn+l>• h + n(n+l)v2 [ n-1] - - n n- - n n~ 2(2n+l) !! ~ n~·
The fact that h is irreducible reduces this result to: n-
n h • <vn+l>• nv2 n-1 y<y >~ n- - n nn + 2n+l <y >n~l h.
n-
(C-10)
(C-11)
This relation has been employed to derive the following useful
result:
V ( h(v)*<vl\) v n- n -
a h <vn+l>- .!_ ....!!_- + <vn-1> • _1_ l..(v2n+l h(v))
- n v av - n-1 2n av n- • v (C-12)
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The relation between the harmonic tensors and Legendre
polynomials can be inferred from the following formula:
( l)n n+l an 1 P (cose) = - v - (-),
n.. n! avn v z
where v2 = v2 + v2 + v2 and v = vcose. x y z z
Comparing (C-1) and (C-14) one obtains:
n!vn <v°> = (2 l)il P (cose). - zzz ••. z n- n
n
(C-13)
(C-14)
Let u be a vector in the direction of the z-axis; the following·
result is then readily obtained:
n nlunvn <!°>n<~°> = <v°>~~ = (2n-l)!! Pn(cose).
Finally a projection operator ffe is defined by: n
JJ. := (2n+l) I! n 411v2nn!
(C-15)
(C-16)
whih then permits the following notation for the orthogonality
property (C-3):
ffJ. h(v)-<v°>=S [h(v)]. mn- n- nmm- (C-17)
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APPENDIX D
The Landau collision integral for identical particles
D-1 The Landau collision integral
The formulation of a kinetic equation for a plasma is rnore
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 5 collision integral reads:
J~ 0(f~,f 0 ) == C 'iJ •JG•(l 'iJ _ l_ 'iJ )f (v)f (v')d3v• ~µ ~ µ a!3 v = m v m v' u !3 '
a !3 (D-1)
q2q2lnA g2! - .s.s where: c = a !3
~ ~ v'- v. a!3 811e: 2m g3
o a
(D-2)
The following properties of ~ will be often used:
~·y = ~·y' , ~ = 'i/v'i/vg. (D-3)
If the distribution functions in (D-1) are isotropic the Landau
collision integral reduces to:
(lf
f - 13 }v' 2dv'. (D-4) a av•
The integration over S\, is done in spherical co-ordinates with
y along the polar axis (y•y'= vv'cos9). Then:
211 1l
/gdOv' = J f {v2 + v' 2 -2vv'cos9}~sin9d6d~ 0 0
-211 I 13
= 3vv' { v-v' - (v+v•)3}.
Two different cases have to be distinguished:
if v'<v : /gdOv' 411(v•2+ 3v 2 ) 3 v •
if v'>v 411 v 2 + 3v' 2 3( v' ).
(D-5)
(D-6)
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Straightforward differentiatlnn of these expressions yields:
v•2 'i/v'i/vfgdf!v' = 41T{"'---<v 2> + y}, if v'<v,
v5 -
=~I , if v'>v. (D-7) 3v'=
Insertion of these results into (D-4) then leads to:
2Caa a m af Jaa = mav2 av [m;fa(v)Io(fa) + J a/i1 2<fa) + J-1Ua)}].
(D-8)
The functionals I and J are defined as: p p
4 v 2 I (f) = ~ J xp+ f(x)dx,
p VP 0
41! ooJ p+2 J = ~ x f(x)dx. p VP v
(D-9)
In the case of identical particles the expression (D-8) can be
written in an elegant way. If $=a (D-8) becomes:
2C a . aln( f ) J = ~z;;: [f (v){I 0(f) + :!:
3 a a (I 2(f) + J_ 1(f ))}].
aa mav av a a v a a (D-lO)
Two of the integrals are evaluated as follows:
f 41Tx2f dx - f 41!x2f dx o a v a
n - 41Tff x 2dx a a '
(D-lla) v
3n kT oo
~ - 41! Jx4 f dx. (D-llb) m v 2 v 2 v a
a
The expression in (D-10) between braces is then equal to:
kT aln(f ) oo aln(f ) n (l+ a a)+ 41ff{-x2 + ~ a (v3-x3)}f (x)dx.
a mav av v 3v av a (D-l 2)
The second term in (D-12) can be written in a more symmetric
form by the observation that:
00 v3 00 3 af 1 00 aln(f ) -Jx2f (x)dx = 3 fa(v) + J ~~adx = - f(x3-v3)f (x) a dx
v a v 3 ax 3 v a ax ,
(D-13)
so that the collision integral when operating on isotropic
functions can be written as follows:
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/
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zc a kT aln(f ) J = ~2 -;;-[f (v){n (1+ __!; av a ) + aa m v ov a a m v
a a
1 3ln(f ) 1 aln(f ) v3)xf (x)(- a a - - a a )dx}].
a x x v v (D-14)
One can now see that this collision integral is zero if f is a a
Maxwellian, i.e. if:
m a
- kT a
(D-15)
The collision integral in (D-14) is still nonlinear. In the
remainder of this appendix the linearized Landau collision
integral will be investigated.
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D-2 The linearized Landau collision operator for like particles
The linearized Landau collision integral for collisions
between identical particles is defined as follows:
J (f) = J (f,f_u) + J (f M,f), aa aa =~ aa a (D-16)
where faM is the Maxwell distribution, see (D-15). If f is
isotropic too, expression (D-10) may be used. The linearized
collision operator then reads: 2c a kT a
J (f) =~-[I0 (f )(1+~-)f(v) + aa m v2 av aM ma.v av
a.
(D-17)
This operator appears in the equation for the isotropic
correction in chapter IV section 4. A generalization of (D-17)
obvious.ly reads: c
J ( f(v)•<v~) = a.av •JG•(V -v )x a.a. n- n - m v = v v'
a
x{fM(v) f(v')•<v•1> - fM(v') f(v)•<v1>}d3v•. n- n - n- n -(D-18)
With the properties of the harmonic tensors, see appendix C,
the first part of the integrand is found to be:
G•(V -v ,)f (v) f(v')o<v'n> = - v v aM n- n-
f(v')J. n-
Concerning the angle integration in (D-18) the following
integrals have to be calculated:
( D-19)
(D-20)
The integral in the right-hand side will still be a harmonic
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tensor, built up by the vector y, so that the following Ansatz
is made:
fg<v'°>do , = H (v,v')<v°>. - v n -
(D-21)
If on both sides of this equation the n-folded inner product n
with <y > is formed the following expression for Hn is
obtained:
v' n H (v v') = (-) fgP (cose)dO ,, n ' v n v (D-22)
where e is the angle between! and v'. The result for n=O was
already obtained in (D-6). After elementary integrations the
result for n=l reads as follows:
v' 4n v•2 H1 = -v fgcose dOv' "'-~v•2- 5v2), if v'< v,
15 v3
4n v2- 5v' 2 "'rr ( v' ) ' if v'> v. (D-23)
For the evaluation of in the case n=l the following expressions
are useful:
Q = {v V (H (v,v')v)•v = i:!!.[v•2(5 9v'2
) 6v'4
] if v'< v, .1 v v 1 - - 15 - -- y + --~ ' v2 v3
4v3 6v2 ] v'~ + v'~ , if v'> v,
Q0 = v v H0(v,v') ,. v v .
4n{(l- v'2
)v + 2v'2r}, if v'< v,
v2 = 3v3 =
o Sn I . f ' ~o = 3v'= , i v > v,
(D-24)
(D-25)
where So has already been derived in (D-7). With these
definitions the linearized operator in (D-18) for n=l reads:
(D-26)
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After some tedious but straightforward calculations this
expression is brought into the following form:
Caa 2v ma 2 2ma Jaa(f(v)•_!) = iil'!•[5(kT ) faMI4(!}- 3vkT faMI2(f> + 161tfaM! +
a a a
kT 1 3 kT 32 + 2I 0(f _,...){(--a + - 2 ~+_a_ -;--2 }f(v)] =: v• 1J (f), (D-27)
w~ m v" v v m v3 QV - - aa -a a
where the symbol J is introduced, see also equation (4-65). 1 aa
Next a change of variables is made:
m v2 a
w = 2kT a
so that for example:
m 3/2 faM • na(2nk~ ) exp(-w),
a
kT 312 I w I I (f) = 4nl2( ma) w-p 2 Jx(p+l) 2f(x)dx.
P - a o
The operator 1Jaa can then be written as follows:
8v lJ Cle/!> • _J!!! w-3 /2e -w.t(f),
rz;
where: n C
\} =~ Cla 2m v 3
a Ta
kT v2 • _...::
Ta m a
(D-28)
(D-29)
(D-30)
(D-31)
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and the linear integro-differential operator is equal to:
1,(f) = j(~S/2 - ~3/2)f(x)dx + (~5/2 - jw312)jf(x)dx + 0 w
2w312f(w) + :w(2wF(w)~!J• (D-32)
One can show that /, becomes a pure differential operator by
differentiating once. After some manipulations it can be cast
into the following form:
-w a r ) a2 [ -2w a2g] a [ -2w !ll] JJg = e awltf = aw2 2wF(w)e p - aw 4F(w)e aw ,
where: a -w
f = a;{e g}. (D-33)
The function F(w) appearing in these expressions was defined in
(D-29) already. An important property is the fo~lowing:
w F(w)e-w = \Jx\e-xdx.
0
F satisfies the following differential problem:
dF \ dw - F = \w , F(O) = O,
from which the following power series valid near w•O is
obtained:
"' F{w) = \ k+3/2 1.. akw , k=O
An asymptotic expansion for w + 00 reads:
1 \ l n-l ( l)kf(k+\) F(x) ~ ~exp(x) - \x - - L - • 4 4/; k=O xk+~
(D-34)
(D-35)
(D-36)
(D-37)
The special interest in the function F stems from its frequent
occurrence in the operators derived from the Landau collision
integral.
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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
p -w'I' q -w A = Jw e ~(w e )dw. pq 0
(D-38)
Straightforward substitution of /, into this expression leads to
the following integrals to be calculated:
00
T = Jwme-wy(n,w)dw, m > -1, mn
0
where y(m,n) denotes the incomplete gamma function:
wf -x n-1 y(m,n) • e x dx, 0
n > O.
The matrix elements A turn out to be: pq
(D-39)
(D-40)
A 2(-5P + -
31
)T ....2. + 2('~l. + l)T 5 - 2(pq+p+q)I(p+q-l) + pq q,p•2 5 3 p,q"'i"
+ r(p+q+3/2) {p+q-\- ~p+q+3/2)(p+q+5/2) t· 2p+q+5/2 5
(D-41)
The function I(k) is defined by: .. I
k__ -2w I(k) := w-F(w)e dw = \Tk,J/2•
0
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
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T = mT + r(m+n) mn m-1,n
2m+n ' T l + T l = r(m)f(n). m- ,n n- ,m (D-43)
From these the following expressions are directly obtained:
T .. r(n)2-n , T - H 2(m+l), O,n m,m+l -n-1
t 1 = 2 (n+2)r(n), ,n
T - (1 -2-n-l)r(n+l), n,l
and so on. For I(k) one easily obtains:
I(k) = kI(k-1) + r(k+3/ 2) 2k+5/2
k > -5/2,
(D-44)
I(O) = t(f )\ , I(\) = ~6 , I(-1) = \1/!!{ln(l+2~) - 2-\}. (D-45)
The matrix elements for the operatorJ are then simply (see
also (5-79)):
'): pq
"" f p-~ q-w w e J!(w e )dw = A.
0 pq
jv(w)wp+qe-Zwdw =
0
1 \ 1T~ .. p+q+2 = Apq - l;"f i;;f(p+q) - 46 fw Q(w)dw
0
(see (4-106) where Q and B were introduced).
The matrix elements for the operator ,'() are defined by: .. o = JwP.t>(wq)dw. pq 0
With the definition (D-33) these are calculated as:
0 pq
which clearly is a simpler expression than (D-41). The
analogous expression with the operator ii reads:
(D-46)
(D-47)
(D-48)
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\ "" + ~$ f(w-p)(w-q)wp+qQ(w)e-wdw, p,q > O.
0
(D-49)
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:
L \ ~ · 1;11 ":! 11 , i I 2 -w )
i-01 = -4- + i-10 = 4ll; +a (w-l)w e Q(w)dw • 0
(D-50)
Finally the matrix elements for the Spitzer operator ~s are
given by:
0
\ {2pq+4(p+q)}1(p+q-1) + rcp+q+3/z) + £..!_r(p+q+1).
2p+q+~ 4 (D-51)
This section is closed with the remark that in the expressions
for the matr.ix 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 litterature6. The recurrence relations
for Tmn and I(k) permit easy calculations of all coefficients.
Page 157
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APPENDIX E
Renormalization 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 se.cond order
contribution is (cf. equation (4-64a)):
f(2) -e,as (E-1)
It is the solution of the following equation:
(E-2)
if an arbitrary isotropic function satisfying the homogeneous
equation is momentarily not taken into account. In higher order
the following equations will appear:
(E-3)
Only solutions proportional to £ are relevant, cf. (E-1), so
that the solution in order n reads:
f(n) (c) -e,as
2n(O) C T (c) i,as ei (1) f(n-l)(c).
m c3 -e,as (E-4)
e
All of these contributions have a singularity at c=O, namely:
(n) It' -3n f • v(c ), c + O, -e,as (E-5)
which relation is valid if f(O) is a Maxwellian and T(l)(c) e,as goes to some constant value if c + O. 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:
Page 158
f(2)norm .. -e,as
00
l f(n) (c) n•Z -e,as
-151-
(E-6)
This expression gives convergent contributions to the transport
coefficients and will be used instead of (E-1).
References to the appendices
1. $.Chapman and T.G.Cowling:"The mathematical theory of non-
uniform gases", Cambridge University Press, 1970.
2. T.W.Johnston, J .Hath.Phys. l..(1966)1453.
3. H.Grad, Phys.Fluids _!(1961)696.
4. J.Wilhelm and R.Winkler, Beitr.Plasmaphysik !(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~
Page 159
-] 52-
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 ge'ioniseerd gas
beschrijven.
De vergelijkingen beschrijven alleen elastische botsingsproces
sen. Het doel van het huidige werk is de beschrijving van
transportverschijnselen bij energieen van thermisch niveau en
de berekening van transportcoefficienten. Aangezien gedeel
telijk geloniseerde gassen in het algemeen plasma's van lage
temperatuur zijn met thermische energieen veel lager dan het
eerste excitatieniveau is de bovengenoemde beperking niet
onrealistisch.
De basisvergelijkingen worden in hoofdstuk II gegeven. Voor de
beschrijving van de Coulomb-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 massaverhouding. Ook de ionisatiegraad wordt aldus
ingeschaald en dit geeft aanleiding tot de indeling van het
gedeeltelijk ge'ioniseerde gas in vier gebieden van zeer zwak
tot sterk geloniseerd.
, In hoofdstuk III wordt het zeer zwak ge'ioniseerde 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
Page 160
-153-
getoniseerde 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 getoniseerde 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-atoom botsingsdoorsnede
in feite zes verschillende isotrope korrektie functies zijn.
Ook verschijnen er nieuwe transport verschijnselen welke op
niet lineaire wijze afhangen van de gradienten en krachten. De
symmetrierelaties van Onsager zijn hiervoor niet meer geldig.
In hoofstuk V wordt het sterk getoniseerde 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
ge!oniseerd 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 transportcoef ficienten.
Numerieke berekeningen in realistische gevallen zijn samengevat
in hoofdstuk VI. De toepasbaarheid van de resultaten in het
zwak geloniseerde gebied hangen sterk af van de gebruikte
electron-atoom botsingsdoorsnede. Voor argon blijkt het
Ramsauer minimum zware beperkingen aan de toepasbaarheid van de
theorie in te houden. Berekeningen van de transportcoefficien
ten worden ook vergeleken met zogenaam.de mengregels. De meng
regel voorgesteld door Frost blijkt, vooral gezien de onnauw
keurigheid waarmee de botsingsdoorsnedes bekend zijn, redelijk
betrouwbaar voor de berekening van transportcoefficienten.
Er wordt ook aandacht geschonken aan zogenaamde seeded
plasma's. Voor een cesium-seeded argon plasma is het electrisch
geleidingsvermogen berekend. Daarbij worden twee sterk van
elkaar verschillende reeksen metingen van de electron-cesium
botsingsdoorsnede vergeleken.
Page 161
-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.1.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 Eindboven op 2 november 1952.
Middelbare schoolopleiding Atheneum-B gevolgd aan bet
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.
Page 162
Stellingen behorende bij het proefschrift van
F.J.F. van Odenhoven
Eindhoven, lS februari 1983.
Page 163
I
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 1 bereikt, maar op
grond van een foutieve berekening. Zij laten namelijk de bijdrage tot de
energief lux van de potentiele energie in het gravitatieveld ten onrechte weg.
I) Landau and Lifshitz: A course of theoretical physics, vol.VI,
Fluid Mechanics, Pergamon Press, 1966, p.398.
II
De eerste orde correctie van bet over een gyratieperiode gemiddelde magne
tische moment van een geladen deeltje in een inhomogeen magnetisch veld is
in het kader vau de adiabatische theorie gelijk aan nul. Dit resultaat
volgt niet uit de berekeningen van Northrop 1
I) T .G.Northrop: "The adiabatic motion of charged particles''•
Interscience Publishers, 1963.
III
In een zeer zwak geioniseerd gas is bet noodzakelijk een isotrope correctie
op de nulde orde verdelingsfunctie van de electronen toe te laten. Een bewe
ring van 8ernstein1 van tegengestelde strekking is derhalve onjuist.
I) LB.Bernstein in: "Advances in Plasma Physics", vol.3, 1969, p. 127.
2) Dit proefschrift, hoofdstuk III.
IV
In een instabiele schuiflaag voldoet de gradient-lengte van bet snelheids
profiel beter als karakteristieke lengte dan de impulsverliesdikte.
v
In de behandeling door de Groot et al.1, van een electron-foton gas is ten
onrechte de dynamische afscherming geheel buiten beschouwing gelaten.
I) S.R. de Groot et al.: "Relativistic kinetic theory", North Holland
Publishing Company,1980.
Page 164
VI
De uitdrukking van Rostoker 1 voor het tensoriele geleidingsvermogen van een
plasma is onjuist. Dit blijkt uit het feit dat zijn uitdrukking niet isotroop
wordt in de limiet: k ~ O. ln een correcte behandeling moet met het inwendige
magnetische veld rekening gehouden worden.
l} N.Rostoker, Nuclear Fusion _!.(1961)101.
VII
De nulde orde verdelingsfunctie van de lichte deeltjessoort in een Lorentz
gas relaxeert naar een willekeurige isotrope functie. De veronderstelling
van Chapman en Cowling1 dat dit een Maxwellverdeling is, volgt niet uit de
Chapman-Enskog theorie
I) Chapman and Cowling: "Tiu! mathematical theory of non-uniform gases",
Cambridge University Press, 1970, p.188.
VIII
De in turbulentie-theorieen vaak gemaakte veronderstelling dat het ensemble
van dynamische systemen uniform is 1, blijkt soms in strijd te zijn met de
dynamics van die systemen2
l} R.C.Davidson: "Methods in nonlinear plasma theory", Academic Press, 1972.
2) l.E.Alber, Proc.R.Soc.Lond. A363(1978)525.
IX
Oplossingen van de electronentemperatuurvergelijking1 duiden er op dat
macroscopische "runaway" van electronen in een gedeeltelijk geioniseerd
gas slechts in uitzonderlijke omstandigheden te verwachten valt.
ll Dit proefschrift, hoofdstuk V.
x
Met betrekking tot de evenwichtige opbouw van onderzoek- en onderwijs
programma 1 s is bet wenselijk om bij de afsluiting van onderzoekcontracten
met bedri.iven een extra percentage in rekening te brengen voor gelieerd
onderzoek van fundamentele aard.
