Kinetic theory of transport processes in partially ionized gases

Kinetic theory of transport processes in partially ionized gases

Citation for published version (APA):

Odenhoven, van, F. J. F. (1983). Kinetic theory of transport processes in partially ionized gases. Technische

Hogeschool Eindhoven. https://doi.org/10.6100/IR82292

DOI:

10.6100/IR82292

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Published: 01/01/1983

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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

PROF.OR.IR. P.P.J.M. SCHRAM

EN

wees slechts bevreesd voor het blijven staan

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

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

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.

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.

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 Water 10. 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

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.

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.

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, s - - n w (r,t) s-s -

=

fvf (r,v,t)d3v, s -Im c c f (r,v,t)d 3v, gs(E,t)

=

f\m2c c f (r,v,t)d3v,

-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 s-s F (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,

a result that can also be obtained directly from equation (2-1) with the velocity function ~m c2. Another quantity of

impor-s impor-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 _{st s} (f ,ft) - 2fd3_td3_{go(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

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:

q2q2lnA s t

Cs t = _8..;;.11_e:..;..2m--o s

(2-13)

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 r

0:

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 cutt-of this expression there appears an integral over the interaction distance diverging at zero and infinity. The approximation made is that one introduces the lenghts r

1 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.

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 funcdistribu-tions 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

~ _at

+

e;v•Vf - e; -_{- e m} •V f _{v e} - wce(~x!?) •Vvfe • eJ _ee

+

J _ea

+

e:J • , _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

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 .•

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/e2 etc. The t_{2 time scale}

-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 0,11, •• )

+ •..

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

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.

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 e2 • The influence of the electron-electron collisions on the electron distribution function is nevertheless large. The form of the zeroth order electron distribution function is shown to be governed by a non-linear integro-differential equation. The asymptotic form of this equation describes the competition between a Davydov and a Maxwell distribution function.

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),

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 T

0 - e me ce-a - e ce- - c e

(3-12)

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

+

_{w c • bx'J f}

(

(1) ) 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 r₀+ ""• the first part in the right-hand side would

increase without bounds with t

0 except if it is demanded that:

ilf(O) eE'

_e,as+ c•'Jf(O) _...:.. •'J f(O)

<lt₁ - e,as me c e,as /O)(/l) ) + ea e,as w ce-c•(bx'J f(l) ) • - 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) e,as (c)

+

-e,as f(l) (c)oc -

+

=e,as f(l) (c) -<cc> -- + •••• (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)

From the right-hand side of (3-18a) i t 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:

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 ce-c•(bxV f(Z) ) - 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) c g(l) "'::: ( )',- 1- • " '~f e,as' ( 1 )

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 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 c2

e

(3-28)

Insertion of expression (3-26a) for f(2) and using appendix A -e,as

then gives:

....!..JP•{c3 T

~r,l

•A'f(l)} + 3c - (l)=(l) - e,as m c

a

Equation (3-29) may be integrated over T_{1 •} 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.

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 ee

-dw

+

e:mene~e·(~ax~)

+

e:

3

mene~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

_e =

1

₂

kT _e

+

~m

_{e e} u2. (3-34)

The macroscopic quantities are also expanded in powers of e: and the multiple time scale formalism (MTS) is applied. From the above balance equations the following equations are obtained in .zeroth orde:t of e::

(3-35)

(3-36) In first order one obtains:

+

en (O)E' •u (0)

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,

h l .. h l

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'T

2

oT

1 e,as -e,as

au(l)

(0) ( -e,as + (2) b) + V•u(l) + en(l) E' +

m n _{e e,as aT}- wceYe asx-

"-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

"QT

₂ aT

-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)

I t 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 equasolu-tion (3-29). Such condi·solu-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 variaequa-tions with T

1, so that the t1

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:

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 -a -e,as

+

} l ) ) •V. 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) e,as o f (c) '

(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) _e,as solely. With this assumption a

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

_e,as

+

n<I>.vv (O) _

.!.

(l)·E''Vl ( (0) ) = ₀

<lT₂ ~ • ne,as e g ·- n ne,as ' (3-57) where the neutral component has been assumed to be homogeneous

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 e,as=q D(l)) '

e kT(O) m c2 af (O)

(3-58)

with: __ _3m e_.,..;..a..;..s_

f(

_{2kT(O) -}e .

₂

SJ

_CT(l)~(l)ac

-1

e,asd3 _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) )

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 ot₁ -e,as -a -e,as ~e,as· ~a dt

(2) (2) (2) (1)

+ V•P _=e,as + en _e,as-E + m n _{e e,as ce -e,as -a -}w (u

+

w )xb

=

(3-65)

at<

0_>

_aE<

2_>

n(O) (-e,as

+ .

e,as + u(2) •V[/O) ) + V ( (2) + E(O) (2) ) e,as at₃

l t

₁ -e,as e,as • ge,as -e,as·Ye,as

(0) (2) (0) (2)

+ en _{e,as-e,as -}u •E' + m n _{e e,as ce-e,as -a -}w u •(w xb) 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,ase,as

-17 (0)

Pe,as (3-69)

3u(Z)

_:.e,as = O. 3'£1

Equation (3-65) may therefore be written as follows: Du(l) m n(O) (_:.e,as e e,as DT₂ Dw +_:.a) + V•P(2) Dt =e,as m2c2 '·T ~ "'a'(l)

a

f

e

{1- _ __,_

_ _,__

-(c'+ 3_{ma'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.

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)

This is the asymptotic solution of (3-72) _{when 12}+ 00 , and is

known as the Davydov distribution function6 • 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 y

₀(c)$2(c)c2dc 0 where y₀(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 c2y₀:

f

_{y 0}c2$ $ 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)c2_{dc = 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:

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),

with: a

n ff0(c)$n(c)c2dc,

0

if the eigenfunctions are orthonormal:

..

f4> 2(c)y0(c)c2dc = 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.

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)}

+

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 c2 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)w312dw

=

~

,

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)

w B(w')+v

- J(

*

ee }dw'

+

C,

o B(w' )T /TA _a +' v _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.

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.

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.

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 T

0+ 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:

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 ) = -a a ~a =a -a £ 2!\m a v2J (f f.)d3v 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

=

h l

=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)

f(l)

a,as = (a l

+

-2 -a •v

+

a v 2)f(O) 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) ).

at

₁ 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)