ELECTRON DISTRIBUTION FUNCTION AND IONIZATION IN SPACE-DEPENDENT PLASMA THEORY

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ELECTRON DISTRIBUTION FUNCTION AND IONIZATION IN SPACE-DEPENDENT PLASMA

THEORY

V. Boﬀi, V. Molinari, G. Spiga

To cite this version:

V. Boﬀi, V. Molinari, G. Spiga. ELECTRON DISTRIBUTION FUNCTION AND IONIZATION IN SPACE-DEPENDENT PLASMA THEORY. Journal de Physique Colloques, 1979, 40 (C7), pp.C7- 535-C7-536. �10.1051/jphyscol:19797259�. �jpa-00219244�

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JOURNAL DE PHYSIQUE CoZZoque C7, suppZSment au n07, Tome 40, J u i l l e t 1979, page C7- 535

ELECTRON DISTRIBUTION FUNCTION AND IONIZATION IN SPACE-DEPENDENT PLASMA THEORY

V.C. Boffi, V.G. Molinari and G. Spiga.

Laboratorio d i Ingeneria NucZeare deZZa Lhziversitd d i BoZogna, BoZogna, I t a l y .

The application of a unidirectional electric field to a gas is one of the most common method for plasma production. However, also in the simplest cases, the agreement between calculations and experimental results is not yet completely satisfactory. h order to improve the situation a better undertanding of all the various phenomena involved by the ionization growth would be required. This work is aimed at analyzing how a nonmaxwellian electron velocity distribution affects the ionization processes. On the other hand, it can be emphasized that the micro- physical knowledge of the kinetic behaviour of electrons is a necessary basis for any progress in plasm physics

.

For this purpuse we start with the integral form of the stationary Boltzmann

that governs the distribution function f (?T,T)of electrons interacting with an infinite homogeneous distribution of neutral particles,

namely z

f(2,~);

J

^{dx e-}

dv(tt)'lv(t "1

^dtf IQ&(t) ,Y(t)] + 0

In Eq.(l) Q is the external source, the collision frequency vz(v) =SrvZr (v) (r=S,R, I) and the kernel k(T1- 7) = ks(T&T)+ V I kI fif-+T), the subscri pts S,il,I standing for scattering, recombination and ionization, respectively, and VI being the mean number of electrons produced by ionization.

The function

x(t)

and T(t) are the solutions to the electron motion equation, and, in general, are given by

- ~ ( ~ ) = ? T - T ( ~ ) ; ~ ( % ) = T - V ( ~ ) , V ( ~ ) = I V ( % ) ] , (2)

- X ( t ) and V(z ) depending on the external field.

Taking a threedimensional two-sided Laplace transform with respect to

x

of both sides of Eq.(l) Wldsthe integral equation

N

_ _

^-Pox^-^{- -} ^A^-^-

f(p,v)= J e f(x,v)dx = P(p,v)+

J

C(5,5t+7).

R3 R3

-

?(E,Tt) di??

,

⁽³⁾

where f$ and

2

are the results of the application of the operator

N - -

to Q(p,v(t)) and k(7'47 (2))

,

respectively.

If k is a kernel of finite rank n, namely

then we get

A

where

4

is the application of the operator, Eq.(3a), to

4l fie,,,

^{m d}

If we define

- - A _ - -

A~(F)=J~~(T)+~(P,V)~~; B~(FI=

Jq1(-

VIQ(P,V)~V,

0)

R3 R3

the coefficients are solutions to the parametri- zed algebraic system

N - -

The solution f (p,v) then follows from Eq. (5) as

where

2 -

-1 is the inverse of the matrix

The problem is thus solved, requiring only the evaluation of the integrals in Eq.(7) according to

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797259

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a suitable choice of Q, and of the two sets of linearly independent functions \)' (GI) and (7)

@=l,. transform f

. .

^,n)

- ^.

(5,T) In order to get the original f must finally inverted. Then the (F,?) ;),its

A

singularities of ~(F,v)

,

of

4

(iS,i?)

,

of the vector -1 - 1

B(F) and of the matrix

2

(p) must be studied. A -

particular role will beeplayed by those values of

;?i

which are roots of det

2(3=O. -

An explicit application of the theory proposed above has been processed for a monodimensional case rbaracterized by a constant electric field, and by a simple model for collisions. This case has been choosen with the aim at exploring the existence of poles for

2 -

-1 (p). If a simple pole, the contribution of which will dominate asymptotically, exists,then it can be referred to as the first ~owf-sedionization coefficient for the physical situation considered.

@or a different model of scattering and ionization, see Ref. 5).

References

1. RIITSCHER,A.: Progress in Electron Kinetics of Lau Pressure Discharges and Related Phenomena, XI11 ICPIG, Invited Lecture, 269, Berlin 1977.

2. BOFFI ,V.C. and MOLINARI ,V.G. : I1 Nuovo Cimento, 34B, 345 (1976).

-

3. BOFFI,V.C., MOLINARI V.G. and WONNEBERGER,W.: I1 Nuovo Cimento,

e,

109 (1978).

4. BOFFI,V.C., kOLINAR1,V.G.: I1 Nuovo Cimento,

G,

77 (1979).

5. BOFFI, v.c., ~ L I N A R I ,v.G. and SPIGA,G. : 11th.

R.G.D. Symposium, Cannes 1978.