A NONLINEAR KINETIC THEORY FOR HF PLASMA GENERATION AND ELECTRON HEATING.

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A NONLINEAR KINETIC THEORY FOR HF PLASMA GENERATION AND ELECTRON

HEATING.

G. Cicconi, V. Molinari, L . Pollachini

To cite this version:

G. Cicconi, V. Molinari, L . Pollachini. A NONLINEAR KINETIC THEORY FOR HF PLASMA

GENERATION AND ELECTRON HEATING.. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-

193-C7-194. �10.1051/jphyscol:1979795�. �jpa-00219501�

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JOURNAL DE. FHYSIQUE

A NONLINEAR KINETIC~THEORY FOR HF PLASMA GEMERATION AND ELECTRON HEATING.

G . Cicconi

*,.v.

~ o l i n a r f * , L. ~ollachini***.

s)tEZectricaZ E n g i n e e r i n g Dept. U n i v e r s i t y o f )t Gerzoa, I t a Z y . jcsaNueZear E h g i n e e r i n g Lab., U n i v e r s i t y o f BOZO@, j I t a l y .

C.I.S.E., Milan I t a l y .

Nowdays a consistent progress has been obtained in describing, by means of kinetic models, the electron kinetics of many gas discharges and plasma generation also in nonsteady state and inhomogeneous to un- derstand the microphysics of the phenomena involved in these plasmas /1//2/. Particularly a nonlinear a- symptotic analysis of the positive column / 4 / and the more extensive theoretical investigations on plasma column at high degree of ionization and under free fall and ambipolar diffusion regimes 151, have been considered. The replacement of the positive column of a glow discharge, by means of a plasma column produced by a HF field, was recently also proved 131.

The theory presented in this paper may be considered in this coptext as a first approach for a more det.

ailed description of plasma generation and electron heating in HF discharges.

We consider a cylindrical column of ionized gas with an alternating electric field, E,(?,

-.

t), at a fre-

quency

,

applied to the whole column symmetrical- ly and directed along the axis of the cylinder. This situation could be regarded, for instance, as an i- dealized model of a plasma column placed in a cylindrical waveguide or resonant cavity. The presence of col~isions permits the transfer of energy from the electric power to the electron gas as an effective thermalization of the energy added to the gas with a subsequent growth of ionization and plasma produc- tion. The aim of this paper is to determine the electron density and electron temperature distributions inside the column for the plasma generated by a HF field, as before sketched, when the global collision frequency of momentum transfer between electrons and other particles,

vC ,

of collisional plasma sa- tisfies the condition g= )%O

.

The plasma is described by the followfng system of

nonlinear equations (for sake of simplicity we consider an infinite plane slab as a valid approximat- ion of a large radius cylinder):

.

-

where the unknowns

r*( ,POI,

(current densities),

Hs, Nt

(total energy flows), ne, T (electron density and temterature) and E,(external electric field) a- re averaged in a HF period. The coefficients

, &ks,

p011i5nb)%TaG()

3%

^m8%

Tlf- 4 n e h Imt Z~YS),

(valid for c*4<Qe ), L Z ~ / ~ , and

the relative velocity. The collision t e n M for e-

C

lectron-neutral and electron-ion collision is con-

sidered as

-

^CI

&i

the ionization potential and

QE/=.

In order to solve the system (11, for a slab of wi- dth 2a, wk assume for n(~) a distribution of the

type,

- t ~ b x s a ,

2 t

- 4

L>&)L

4 ,

+I=

^me

⁽⁴ - ^{~ 6 0} ^x ^/a ^1,

where ).(y,, is an unknown function. By simple al- gebra an equation, only dependent o n >

,

^{for the}

electron temperature can be obtained. Moreover it is possible to prove that if the central zone of the slab is only considered, we have

A *

^const

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979795

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s i n c e i s a v e r y smooth function. By t a k i n g i n t o ac- count t h i s hypothesis, w i t h boundary c o n d i t i o n s

fa

10) T-

^t

(66/dr), * 0,

t h e p a r a m e t r i c e q u a t i o n f o r t h e e l e c t r o n temperature f o r small v a l u e s of x / a , becomes

where >z ( ~ and

&(+'Cb&v

m^=;onst*

For t h e e l e c t r i c f i e l d we o b t a i n t h e s o l u t i o n ,

where F ( a , b ; t ) i s Gauss' c o n f l u e n t hypergeometric

E o ( a ) i s a c o n s t a n t w h i c h s a t i s f i e s t h e boundary con- d i t i o n s on t h e s l a b s u r f a c e . For small v a l u e s of x / a t h e ( 4 ) simply becomes

which i s n o t dependent of! %

.

The e q u a t i o n f o r

3

i s t h e following:

A d i f f u s i o n l e n g t h may be d e f i n e d as a f u n c t i o n of "

a s follows

w= **^(*O/hk\)i** = & / i n ,

I n Fig. 1 a r e shown ;he c a l c u l a t i o n r e s u l t s o f

I A ~

v e r s u s E, i n t h e c a s e of d i f f u s i o n dominated plasma (DDP) f o r T = 0 and w i t h T a s a parameter.

s eo

I n Fig. 2 f o r a recombination dominated plasma (RDP) o r i o n i z a t i o n dominated plasma (IDP) a r e shown t h e v a l u e s of

\/\I

v e r s u s

IT

/ w i t h Teo a s a parameter.

I n Fig. 3 t h e d i s t r i b u t i o n o f T a l o n g /x/ i s shown e

a s a f u n c t i o n of T /T f o r = 1 and (X = -1.

s eo

We c a n conclude w i t h t h e following statements:

1 ) The g r a d i e n t s of n ( x ) and T ( x ) a r e o p p o s i t e and a r e dependent on t h e s o u r c e term T

.

I n DDP and IDP

& s e s (T ) 0 ,

3 >

^{0 )}

^,

on t h e c e n t r e of t h e column:

s

no= max and T = min. I n BDP c a s e (T$ O,X(O), eo

no= min and T = max.

eo

2) The r e s u l t s a r e c o n s i s t e n t w i t h t h o s e r e p o r t e d

i n / 4 / / 5 / and w i t h t h o s e of t h e conventPona1 theo- r y of t h e p o s i t i v e column.

References.

1 ) A. Rutscher. I n v i t e d l e c t u r e of X I 1 1 ICPIG, Ber- lin,1977, 269.

2) T. RGtitka and K. Rohlena. I n v i t e d l e c t u r e of X I ICPIG, Praha, 1973, 61.

3 ) M. Moisan, C. Beaudry and P. Leprince. IEEE Trans.

on Plasma Science. PS-3, 1975, 55.

4 ) H. W. Friedman. The P h y s i c s of F l u i d s . 1967, 2053.