A parametric study of electron energy distribution ... - Springer Link

Plasma Chemistry and Plasma Processing, Vol. 12, No. 3, 1992

A Parametric Study of Electron Energy Distribution Functions and Rate and Transport Coefficients in Nonequilibrium Helium P l a s m a s (3. Capriati, t (3. Colonna, t C. Gorse, | and M. Capitelli I Received August 2, 1991; revised November 19, 1991

Electron energy distribution ./i~nctions in helium plasmas have been calculated by solving the Boltzmann equation at given values of reduced electric .field, in the presence of superelastic and electron-electron collisions. Analytical expressions have been found connecting macroscopic coefficients to reduced electricfield E/N, relative metastable concentration [He(2~S)]/N, and degree of ionization n,./N. KEY W O R D S : Boltzmann equation; superelastic collisions; electron-electron collisions; transport and rate coefficients.

I. I N T R O D U C T I O N H e l i u m is often used in p l a s m a t e c h n o l o g y a n d is an i m p o r t a n t gas c o m p o n e n t o f p l a s m a mixtures in a great n u m b e r o f a p p l i c a t i o n s . T h e r e f o r e n u m e r o u s efforts have been m a d e t o w a r d u n d e r s t a n d i n g the p r o p e r t i e s o f h e l i u m p l a s m a s . In p a r t i c u l a r , s o m e studies have strongly e m p h a s i z e d the n o n - B o l t z m a n n c h a r a c t e r o f the electron energy d i s t r i b u t i o n function (e.e.d.f.). In a p r e v i o u s work ~'~ we s h o w e d the i m p o r t a n c e o f s u p e r e l a s t i c elect r o n i c collisions from m e t a s t a b l e He(23S) a t o m s in affecting the e.e.d.f.: we p r o v e d that the p r e s e n c e o f fixed c o n c e n t r a t i o n s o f He(2~S) p r o d u c e s a long p l a t e a u in the tail o f the e.e.d.f, as a result o f s u p e r e l a s t i c and elastic collisions, an effect which is m o r e i m p o r t a n t the lower the r e d u c e d electric field E / N a n d the greater the m e t a s t a b l e c o n c e n t r a t i o n [He(2-~S)], a strong effect b e i n g o b s e r v e d for relative m e t a s t a b l e c o n c e n t r a t i o n s greater than 10 -6 . Centro di Studio per la Chimica dei Plasmi, and Department of Chemistry, University of Bari, Bari, Italy. 237 0272-4324'92/090t)-O237$t)6 50 0 ~ ' 1992 Plenum

Publishing Corporation

238

Capriati, Colonna, Gorse, and Capitelli

In a later work ~2~we noted that metastable and electron concentrations are generally similar; therefore, at relative metastable concentrations larger than 10 -6, electron-electron ( e - e ) collisions should be taken into account. Therefore, in Ref. 2 we studied the competitive effects of superelastic and e - e collisions on the e.e.d.f.'s in helium. We showed that the presence of e - e collisions tends to cancel the effect produced by superelastic collisions, even though these last processes still dominate at low reduced electric fields ( E / N < 6 Td) in a wide range of ionization degrees (10 ~< n,. < 10 3). The aim of the present work is to extend this kind of analysis to the study of some macroscopic parameters (rate coefficients, electron diffusion coefficient, electron mobility, electron mean energy) and to obtain analytical relations in which such macroscopic parameters in helium plasmas are expressed as a function of reduced electric field, relative metastable concentration, and degree of ionization. These relations may be very useful in modeling cold plasmas by means of fluid models. 2. P H Y S I C A L M O D E L Electron kinetics in cold plasmas can be described by means of the Boltzmann transport equation ( O + v . gradr+ e E . g r a d , ) f ( r , v, t) = (~-t) m cou

(1)

wheref(r, v, t) is the electron distribution function at time t, spatial location r, and with velocity v. Equation ( 1 ) may be solved in the two-term approximationC3'; including momentum transfer, inelastic and superelastic processes, and e-e collisions, it can be expressed in terms of the electron number density in the energy space n ( s ) as ~4'-~

On(s, t ) _ Ot

OJl Os

OJ~.~ OJ,. , . + i n + S u p Os Oe

(2)

where n(e, t) de is the electron density in the energy range e, e + de, (OJt/Oe) is the electron flux in the energy space due to the electric field, (OJ J O e ) is the electron flux in the energy space due to elastic collisions, (OJ,.,./Os) is the electron flux in the energy space due to e-e collisions, In is the electron flux in the energy space due to inelastic collisions (excitation and ionization), and Sup is the electron flux in the energy space due to superelastic collisions. The very simple helium atomic model shown in Fig. 1 was considered. Table l shows the elementary processes inserted in the model (in the Appendix we briefly state the reasons why we chose such processes among all the ones according to the atomic model considered).

Macroscopic Coefficients in Helium

239

e(eV)

/

He+(ls)

24.58

21p 21.21

21

2 tS 20.61

20--

23S

19.82

lg--

0

11S

Fig. !. Helium atomic model taken into account.

3. N U M E R I C A L M O D E L If the f u n c t i o n n(s, t) is put into finite difference form as a vector n~(s, t) = hi(t) (1-< i . 3'

e

aM=O

"

aM=le

4

2000

g u

1500

4.-' 1000 ,-4 ",-4

,m !

500

.

.

.

.

.

,

,01

. . . . .

,

,I

,

10

I

100

E/N (Td) Fig. 5. Electron mobility as a function of r e d u c e d electric field at three different values of the a M p a r a m e t e r (0, 10 "~, 10 7). (e-n m e a n s 10-".)

"l

108

1-

(2

~

I°7 "

v

°

present

work

[8]]

/

[9

4J -,.4 i06 U 0 @ >.

lo 5

.t-,' u,..l i04

.......

,01

,

........

,I

,

........

I

,

I0

........

I00

E/N (Td) Fig. 6 . Electron drift velocity as a function of reduced electric field at a M = 0: c o m p a r i s o n between the present work and some e x p e r i m e n t a l results.

Macroscopic Coefficients in H e l i u m

249

80O0 T

-o- a M = O

0 o

-'- a M = l e - 4 6000

aM=le- 3

S U

4000 0

0 0

2000 ,

v

,

: :::

:

. . . . . . . .

I

,Ol

---

.....

I

,I

. . . . . . . .

.

. . . . . . . .

10

I

100

E/N (Td) Fig. 7. E l e c t r o n d i f f u s i o n coefficient as a f u n c t i o n o f r e d u c e d electric field at t h r e e different values o f the a M p a r a m e t e r (0, 10 -4 , 10 3). ( e - n means 10-".)

16

>

aM=O

(D

/

-'- a M = l e - 4 12 -'- a M = l e - 3

~4

M

,Of

,I

I

I0

100

E/N (Td) Fig. 8. E l e c t r o n m e a n e n e r g y as a f u n c t i o n o f r e d u c e d electric field at t h r e e different values o f the a M p a r a m e t e r (0, 10 -4 , 10-3). ( e - n

means 10-'.)

250


2e+23

-I

O ®

•

present

work

(B

present

work

(A

i / /

[12]

"7 O

Ie+23

v

0e÷0 ,01

,I

1

E/N

10

100

(Td)

Fig. 9. Product of electron diltusion coefficient and gas density as a function of reduced electric field at c~M =0: comparison between the present work and some experimental r e s u l t s . - IA) Present work with energy grid mesh size 0.50 eV; (B) present work with energy grid mesh size 0.02 eV up to 0.6 Td and I).5(IeV beyond 0.6 Td. (e-n means 10 ".) 1. All the m a c r o s c o p i c coefficients d e p e n d i n g on the low-energy part of the e.e.d.f.'s (transport a n d superelastic rate coefficients) are weakly d e p e n d e n t on a M (Figs. 5, 7, 8, a n d 15-16). This is due to the fact that this energy region of the e.e.d.f.'s is practically i n d e p e n d e n t of the collisional processes c o n s i d e r e d (as can be inferred from the Fig. 2). Moreover, Eqs. (8)-(12) reveal that all the m a c r o s c o p i c coefficients have the same d e p e n d e n c e on o~M (c~M v, where p is a real n u m b e r such that 0 . 6 < p < 2 . 6 ) . 2. At low E / N values ( < l T d ) the m a c r o s c o p i c coefficients are i n d e p e n d e n t of E~ N. The reason for this is that at low E~ N values energy exchanges due to elastic and superelastic collisions are d o m i n a n t with respect to those due to e l e c t r o n - e l e c t r i c field interaction, c a u s i n g the matrix C + T(n) terms i n t r o d u c e d by the collisional processes to d o m i n a t e on the terms due to the electric field. 3. At i n t e r m e d i a t e E~ N values (1 < E / N < 10 Td), one or two changes of concavity are generally present. Such changes in the s e c o n d derivative are due to new energy loss processes: inelastic collisions. 4. At high E / N values ( > 1 0 T d ) the m a c r o s c o p i c coefficients are i n d e p e n d e n t of a M . At such E~ N values e l e c t r o n - e l e c t r i c field interactions are d o m i n a n t with respect e - e a n d e - a t o m interactions, c a u s i n g the electric

Macroscopic

Coefficients

in

Helium

251

102 • • "

1o I

- - -

[el [13] [14]

- , o ,~

presentyork(B presentvor~=g~"

I° o

8

io_i

B

i0-? .01

.,

.

,

,I

-

.

,

I

E/N

I0

I00

(Td]

Fig. I0. Electron diffusion coefficient: m o b i l i t y ratio as a function of r e d u c e d electric field at o~M = 0: c o m p a r i s o n b e t w e e n the present work and s o m e e x p e r i m e n t a l r e s u l t s . (A) Present work with e n e r g y grid mesh size 0.50 eV; IB) present work with energy grid mesh size 0.02 eV up to 0.6 Td and 0.50 eV b e y o n d 0.6 Td.

,-4 I

0 tfl

t61~

.......

16 I~

: Z [z~zZ]

16t] --e- a l l . 0

16 I,

U

aM=le--7 ,,'4 A I

f.L10

M

16 I~

ctil= l e . - 6

161i 16 II

- ~ a M,,,le.--4

l(~ It ,01

a li-1~--5 - ' - a Id- l e - - 3 . . . . . . .

,

.......

,I

E/N

,

.

1

,

I0

. . . . . . .

I00

(Td)

Fig. I I . He(2~S) e x c i t a t i o n rate coefficient Ifrom g r o u n d statel as a function of r e d u c e d electric field at ~'M = 0, 10 -7 10 ", 10 ~ 10 a, and 10 3. ( e - n m e a n s 10 ".)

252


1o-7. ,,,4

.............

I

0 0 01

;=:f

-

.-,..,,,

i¢e

o~

io_g •- e - a N , , 0

"*" a !1- l e " 7 C'sl A

a)d-le-6

161(

•-e- a H . l e . - 5

I

"-*" a ] l - le---4 -"- a N-le-3 IO t]

,01

.

.

.

.

.

.

.

.

,

.

.

.

.

.

.

.

,I

w

.

.

.

.

.

.

.

.

1

,

.

.

.

.

.

.

.

I0

I00

E/N (Td) Fig. 12. He(2~S) excitation rate coefficient (from He(2"~S) state) as a function of r e d u c e d electric field at a M = 0, 10 7, 10 ~', 10 -5, 1 0 - 4 , and 10 ~. (e-n m e a n s 10 ".)

IO-9j 7

161] 161]

16~t

161~ -~ aloO

t61~

-I'-a l - l e - 7

-o- a % - 1 o - 6 -*- a l ~ l ~ - 4 16

-*- a1-I¢~-3 ,

,01

. . . . .

,I

E/N

,

-

°

I

,

I0

.

100

(Td)

Fig. 13. H e ( l ~ S ) i o n i z a t i o n rate coefficient as a function of reduced electric field at a M = 0 , 10 -7 , 10 -~', 10 5 10 "*,and 10 -3 . (e-n m e a n s 10-".)


253

Io-6,

to-7~

7

,o~ [61!

Io-),~ [e):I

--e- a{l=O

~e)(] 16q

I pt"

"~ aM=le-7 ""- a N= l e - 6

/

....

162~

" ~ a ~1-1e-5 "0- a {!= le.--4

~

- * - a W= le-3

,

-

-,

-

-

,I

,01

-

,

.

.

.

.

.

.

t

I

.

.

.

.

.

.

.

.

I0

,

I00

E/N (Td) Fig. 14. He(23S) ionization rate coefficient as a function of reduced electric field at o~M=0, 10 7 10 ¢', 10 ~, 10 -4, and 10 ~. ( e - n means I0-".)

2,5a-9"

I

,

0

- -

-

-

,

,

,

2,0e--9-

1,5~-- 9 '-o- a l l . 0

0

-b- a I1= le... 7 o A

1,0e-9

a ~=1e.-6 --e- a i t . 1 @ ' - 5

I

M

aIW=le-4

"-*- a)l- l e - 3 5,0~ I0

.

,01

.

.

.

.

.

.

.

•

,I

.

.

.

,

.

.

.

.

.

.

I

,

I0

.

.

.

.

.

.

.

IO0

E/N (Td) Fig. 15. He(23S) superelastic rate coefficient as a ['unction of reduced electric field at o~M = 0,10 ~,10 ,10 ,10-4, a n d l 0 ~ . ( e - n m e a n s l 0 ".)

254


,,-4 I r~

2,0e.-9,

1,5e--9. rO

/

A I

,,0,-9

.o.t

/

/

,I

I

E/N

(Td)

a I1-,1.--7

~

-'-~,~-1.-6 "m- a ~" l e ' - 5

1o

1o(3

Fig. 16. H e ( 2 t P ) superelastic rate coefficient as a function of reduced electric field at ~ A ' ! - 0 . 10 ". 10 ~'. 10 ~. 10 4. and 10 ~. ( e - n means 10 ".)

field terms o f the matrix C + T(n) to d o m i n a t e the terms i n t r o d u c e d by the collisional processes.

4.4. M o b i l i t y The results o b t a i n e d for the electron m o b i l i t y as a function o f r e d u c e d electric field at fixed values o f o~M are shown in Fig. 5. For the sake o f clarity only the curves at o~M = 0 , 10 4, 10 ~ are shown. I n t e r p o l a t i o n o f such results y i e l d e d Eq. (8) with the coefficients shown in Table II. Two o h m i c b e h a v i o r zones are d i s t i n g u i s h a b l e : one at E / N < I T d , a n d the o t h e r at E / N > I O T d . In the range 1 < - E / N < I O T d , inelastic collisional processes cause the m o b i l i t y to d e c r e a s e by 50% on the average. In the first o h m i c zone, electron m o b i l i t y is a d e c r e a s i n g function o f o~M. This is due to the fact that in our m o d e l collisions are isotropic, hence an increase o f collisional frequencies causes a d e c r e a s e o f the e.e.d.E a n i s o t r o p y and o f electron m o b i l i t y (which, as is known, is a sort o f index o f e.e.d.E a n i s o t r o p y ) . The lack o f d e p e n d e n c e on E~ N at values less than 1 Td is due to the smallness o f E~ N, while, at values greater than 10 Td, it is due to a sort o f s a t u r a t i o n r e a c h e d by the collisional processes (the energy thermal redistrib u t i o n is p r o p o r t i o n a l to the electric field energy gain).


255

The reliability o f the results o b t a i n e d was c h e c k e d by c o m p a r i n g the t h e o r e t i c a l drift velocity (w = p.E) to the values m e a s u r e d in some experimental works (Fig. 6). The a g r e e m e n t is satisfactory: /~(cm2s IV I)=D+

A

A, B, D = S - K ( a M ) " ,

1-B(E/N)' I + B( E / N ) '

(8)

c : 1.8

4.5. Electron Diffusion Coefficient and Mean Energy

The results o b t a i n e d for the electron diffusion coefficient a n d the electron m e a n energy as a function o f r e d u c e d electric field at fixed values o f a M are s h o w n in Figs. 7 a n d 8, respectively. Again, for the sake o f clarity, only the curves at o~M = 0, 10 4, a n d 10 ~ are shown. I n t e r p o l a t i o n o f such results y i e l d e d Eqs. (9) and (10) with the coefficients shown in T a b l e s III a n d IV. Both m a c r o s c o p i c quantities are c o n s t a n t up to 1 Td a n d strongly increase b e y o n d 1 Td, a c h a n g e o f c o n c a v i t y being present at nearly 20 Td. This s i m i l a r b e h a v i o r is due to the s i m p l e relation which links the diffusion coefficient a n d the m e a n energy: the m e a n energy is p r o p o r t i o n a l to the c h a r a c t e r i s t i c energy which, in turn, is c o n n e c t e d to the diffusion coefficient by the Einstein relation KT,. = D / # (K is B o l t z m a n n ' s c o n s t a n t and 7",. is the electron t e m p e r a t u r e ) . At low values o f E / N ( < I Td) a lack o f d e p e n d e n c e on E / N can be o b s e r v e d : in such an interval the e.e.d.f, is scarcely d e p e n d e n t on E / N , as p r e v i o u s l y observed. However, o u r n u m e r i c a l m o d e l overestimates the diffusion coefficient a n d the m e a n energy, when a M = 0. At low values o f E / N the electric field is not a source o f energy; therefore the electrons t h e r m a l i z e with the heavy species. If the energy grid mesh size (Ae) is greater than the gas t e m p e r a t u r e , the electron m e a n energy will be influenced only by such a mesh size and will be greater than the gas t e m p e r a t u r e . An o v e r e s t i m a t i o n o f electron diffusion coefficient will also result. Such c o n s i d e r a t i o n s are p r o v e d in Figs. 9 a n d 10 where o u r theoretical D - N a n d D / a values are c o m p a r e d with some e x p e r i m e n t a l ones: for curve A ( A e = 0 . 5 e V ) the a g r e e m e n t may be c o n s i d e r e d satisfactory e v e r y w h e r e except at E~ N < 0.6 Td, where o u r theoretical values are greater than the e x p e r i m e n t a l ones; r e d u c i n g A t ( A e = 0 . 0 2 e V ) leads to better a g r e e m e n t with the e x p e r i m e n t a l results (curve B). Beyond 1 Td the onset o f inelastic collisions (sources o f isotropy) occurs and c o n s e q u e n t l y the diffusion coefficient is e n h a n c e d . The m e a n


256

energy i n c r e a s e s b e c a u s e the electric field increases. D(cm2s-')

=

g (eV) =

[

) E / N -20Td

i=0

c,+b,(aM) .](E/N),

6.08 + 0.067(

El N),

if

) E / N < 20 Td[

ifE/N>-20Td

J (9)

(10)

J

4.6. Inelastic Rate Coefficients

S o m e o f the results o b t a i n e d for inelastic rate coefficients as a function o f r e d u c e d electric field at fixed values o f c~M are s h o w n in Figs. 11-14 where only K ~El K F1~2, K~ a n d K ~ E a n d K E are i are r e p o r t e d ( K ~.2 s i m i l a r to K ~ . , KV_~ a n d K2_ E 3 are s i m i l a r to K~-2; K;~ a n d K~ are similar to K I ) . I n t e r p o l a t i o n o f such results y i e l d e d Eq. ( l l a ) with the coefficients s h o w n in Tables Va, b, c. In such a case it was p o s s i b l e to s e p a r a t e K(E/N, aM) in two parts, one d e p e n d e n t only on a M , a n d the o t h e r d e p e n d e n t only on E/N, i.e., K ( E~ IV, o~M) = K ' ( a M ) + K"( E~ N ), such that K ' is d o m i n a n t at low E~ N values a n d K" at high E~ N values. At low El N values ( < 1 Td) the inelastic rate coefficients are essentially d e t e r m i n e d by the e.e.d.f, p l a t e a u d u e to s u p e r e l a s t i c collisions. Such a situation causes the inelastic rate coefficients to be i n d e p e n d e n t o f E / N a n d d e p e n d e n t on a M . The situation is reversed at high El N values. A different a n d s i m p l e r kind o f i n t e r p o l a t i o n was necessary for the E e x c i t a t i o n rate coefficients K 2~3, ~ K ~ 4 , and K 3~4 b e y o n d a t h r e s h o l d value (E/N)* o f the r e d u c e d electric field: Eq. (1 l b ) with the coefficients shown in T a b l e Vd. "

"

K (cm3s ' ) = c ( a M ) P + e x p f - e * ~ (E/N)i a; 1 +c* i:o c(M, a/M) K(cm3s

= co[l + 4 0 0 . 5 A~ ')=,~=o(E/N)'

M"7(

(lla)

-x/~/M)] (lib)

4.7. Superelastic Rate Coefficients

Some o f the results o b t a i n e d for s u p e r e l a s t i c rate coefficients as a function o f r e d u c e d electric field at fixed values o f a M are s h o w n in


257

Figs. 15 and 16, where only K s~ o and g ~so are reported (K 240, s K 2s ~ , S S S K3~t are similar to K~_o, whereas K 3-2 is similar to K3~o). Interpolation of such results yielded Eq. (12) with the coefficients shown in Tables Via, b. Firstly, in Figs. 15 and 16 one observes an opposite trend between the two superelastic rate coefficients. Secondly, a total lack of dependence on the a M parameter is seen (see "General features"). At low values of E / N (20 Td) the electric field is the dominant source of energy; therefore, superelastic rate coefficients become constant again. The different behavior of the superelastic rate coefficients can be understood by looking at Fig. 17 where the quantity R(e)= (e/2m,,)~/2o-(e) for the superelastic processes with rate coefficients K ~s o , K ~s o , and K 3s~ o is plotted as a function of energy. It can be seen that R(e) strictly decreases for the first two superelastic processes while it increases for the third one; therefore, as the maximum of the e.e.d.f.'s moves toward higher energies when the electric field increases, the first two rate coefficients decrease when the third one increase: (12)

5. C O N C L U S I O N In the present work we have calculated the macroscopic coefficients (electron mobility, electron diffusion coefficient, electron mean energy) for helium plasmas at a gas density of 2.45 × 10 ~ c m 3 and at a gas temperature of 300 K; in addition, the ionization rate coefficients from the He(IIS), He(2'S), He(2~S), and He(2~P) states, and different kinds of excitation rate coefficients with corresponding superelastic ones have been derived. Analytical expressions connecting all the above macroscopic coefficients to the reduced electric field, the relative He(23S) metastable concentration, and the degree of ionization have been determined.


259

,o°l 10"~

16 4

i0 -6 i

> h

108L

UJ LU

10~°_

1~ 2 _

10 TM _ I

-~! 10 0

I 10

I 20 ENERGY

1 30

40

50

(eV)

Fig. 18. E.e.d.f.'s at r e d u c e d electric field = 2 T d , d e g r e e o f i o n i z a t i o n = I0 -a, [He(23SJ]/[He(I'S)]=IO a [He(2,S)]/[He(l~S)]=lO ~, a n d

[He(2*P)]/[He(I*S)]= 10 listed in the appendix: - group 4.

" in the presence of the groups of processes group I; . . . . group 2; - - - group 3; . . . . .

The accuracy of our model has been tested by comparing our theoretical v a l u e s f o r D. N, D/Ix, a n d w, w i t h s o m e e x p e r i m e n t a l r e s u l t s . (s-'4~ The present parametric study of transport properties may be highly u s e f u l in d i m e n s i o n a l r f d i s c h a r g e m o d e l s u s i n g t h e l o c a l e q u i l i b r i u m a p p r o x i m a t i o n . W o r k is in p r o g r e s s to e x t e n d t h e s e k i n d s o f r e s u l t s to molecular gases.

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6. A P P E N D I X The p r o b l e m c o n c e r n i n g the choice o f the p r o c e s s e s to be taken into a c c o u n t was s o l v e d in the f o l l o w i n g way. F o u r g r o u p s o f p r o c e s s e s were c o n s i d e r e d (Table VII), with cross sections t a k e n from Re(. 15. E q u a t i o n (2) was solved f o u r times, each time t a k i n g into a c c o u n t only one o f the p r e v i o u s g r o u p s o f processes, u n d e r the f o l l o w i n g c o n d i t i o n s : - - g a s d e n s i t y = 2.45 x 10 ~ c m ~, - - g a s t e m p e r a t u r e = 300 K, - - r e d u c e d electric field = 2 Td, --[He(fS)]/[He(I'S)] = 10 4 --[He(2~S)]/[He(I'S)]= 10 5, --[He(2'P)]/[He(I'S)] = 10 -6 , - - n , . / [ H e ( I~S)] = 10 -4. Figure 18 shows the e.e.d.f.'s o b t a i n e d . It can be o b s e r v e d that such curves do not differ significantly; t h e r e f o r e the s i m p l e s t m o d e l ( g r o u p 1) was chosen. ACKNOWLEDGMENTS This w o r k was p a r t i a l l y s u p p o r t e d by C N R t h r o u g h " P r o g e t t o finalizzato sistemi i n f o r m a t i c i e c a l c o l o p a r a l l e l o : s o t t o p r o g e t t o scientifico p e r g r a n d i sistemi." REFERENCES I. C. Gorse, J. Bretagne, and M. Capitelli, Phys. Lett. A 126, 277 (1988). 2. G. Capriati, C. Gorse, and M. Capitelli, Proceedings of 10th ESCAMPIG, Orleans (F), August 28-31, 1990. 3. T. Holstein, Phys. Ret,. 70, 367 (1946). 4. S. D. Rockwood, Phys. Rev. A 8, 2348 (1973). 5. C. J. Elliot and A. E. Greene, J. Appl. Phys. 47, 2946 (1976). 6. R. K. Nesbet, Phys. Rev. A 20, 58 (1979). 7. J. J. Lowke, A. V. Phelps, and B. W. Irwin, J. Appl. Phys. 44, 4664 (1973). 8. R. W. Crompton, M. T. Elford, and R. L. Jory, Aust. J. Phys. 20, 369 (1967). 9. H. B. Millov and R. W. Crompton, Phys. Rev. A 15, 1847 (1977). 10. A. V. Phelps, J. L. Pack, and S. Frost, Phys. Rev. 117, 470 (1960). 11. R. A. Stern, Proc. 6th Int. Conf. Phenomena in Ionized Gases, Paris, 1963. 12. G. Cavalleri, Phys. Rev. 179, 186 (1969). 13. S. A. J. AI-Amin and J. Lucas, J. Phys. D 20, 1590 (1987). 14. H. N. Kucukarpaci, H. T. Saeler, and J. Lucas, J. Phys. D 14, 9 (1981). 15. S. Daviaud, -Caractdrisation dnergdtique et spectroscopique d'une decharge microonde d'helium en flux," Universitd de Paris-Sud, Centre d'Orsay, These, 28 Juin 1989.