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Electron-ion interaction in a weakly ionized gas
Volume 38A, n u m b e r 2
ELECTRON-ION
PHYSICS
LETTERS
INTERACTION
IN
17 J a n u a r y 1972
A WEAKLY
IONIZED
GAS
*
J . - C . G A U T H I E R a n d C. SOL Institut d'Electronique Fondarnentale, Laboratoire associ~ au C. N. R. S., Universitb Paris-XI, 91-Orsay, France Received 22 November 1971
The effective e l e c t r o n - i o n collision frequency has been m e a s u r e d in a weakly ionized gas. Results a r e in good a g r e e m e n t with c l a s s i c a l t h e o r i e s for the e l e c t r o n - i o n interaction in p l a s m a s .
Recent computations [1,2] show that the high frequency conductivity ratio including electrone l e c t r o n i n t e r a c t i o n s c a n b e d i r e c t l y u s e d to d e t e r m i n e the e l e c t r o n ion c o l l i s i o n f r e q u e n c y if t h e r a t i o of t h e e l e c t r o n n e u t r a l to e l e c t r o n ion c o l l i s i o n f r e q u e n c y i s g r e a t e r t h a n 10 -1. O u r t h e o r y a n d S p i t z e r ' s c l a s s i c a l t h e o r y [3] a r e in g o o d a g r e e m e n t , in t h e i r c o m m o n r a n g e of v a lidity (weakly interacting plasmas), although they show a fairly large discrepancy with older e x p e r i m e n t a l r e s u l t s [4, 5]. H o f f m a n n a n d S k a r s g a r d h a v e p u b l i s h e d r e s u l t s [6], u s i n g c a v ity t e c h n i q u e s , w h i c h s e e m to i n d i c a t e good agreement with Spitzer's theory over a limited r a n g e of e l e c t r o n d e n s i t i e s ((1 - 5) x 1010 c m - 3 ) . W e r e p o r t h e r e m i c r o w a v e m e a s u r e m e n t s of t h e electron-ion collision frequency for momentum t r a n s f e r in a r o o m t e m p e r a t u r e a r g o n a f t e r g l o w a t a p r e s s u r e of 5 t o r r w h i c h a r e in good a g r e e m e n t w i t h o u r t h e o r y in t h e r a n g e 4 x 109 8 x 1 0 1 0 c m - 3 . M e a s u r e m e n t s w e r e m a d e 800 to 1200 ~ s a f t e r t h e e n d of t h e a c t i v e d i s c h a r g e . The plasma is contained inside a cylindrical g l a s s b o t t l e of 40 c m l e n g t h a n d 9 m m i n s i d e d i a m e t e r . It i s i n s e r t e d a l o n g t h e a x i s of a standard X-band waveguide; our microwave diagnostic system has been described elsewhere [7-9]. We take into account the residual microw a v e r e f l e c t i o n s a n d l o s s e s of t h e g a s b o t t l e . T h e e l e c t r o n g a s i s in e q u i l i b r i u m w i t h the n e u t r a l s a t r o o m t e m p e r a t u r e (295°K) w i t h i n 10°K, and the high electron thermal conductivity insures a spatially uniform electron temperat u r e [1]. It i s w e l l k n o w n t h a t , in t h e h i g h f r e q u e n c y l i m i t u 2 / w 2 << 1, t h e c o n d u c t i v i t y r a t i o b e t w e e n t h e r e a l p a r t (~r a n d t h e i m a g i n a r y p a r t * This work was supported by D. R. M. E. under cont r a c t no. 70/670.
F
"~ - - ~ - - - ~ ]
]
]
I
I
I --
3
lO 9 _
10 8
• 1010
__ t _ _ ± ~ n e (cm -3)
1011
Fig. 1. Plot of experimental and theoretical electron ion collision frequency at room t e m p e r a t u r e , v e r s u s average density, ~i of t h e h i g h f r e q u e n c y c o m p l e x c o n d u c t i v i t y [10] b e c o m e s ~ r / ~ i = Ueffne/O)ne, w h e r e t h e b a r d e n o t e s s p a t i a l a v e r a g i n g . In a w e a k l y i o n i z e d g a s , it h a s b e e n s h o w n [2] t h a t t h e t o t a l c o l l i s i o n f r e q u e n c y Ueff c a n b e w r i t t e n Ueff = Uei + Veo w h e r e Ueo a n d Uei a r e r e s p e c t i v e l y t h e e l e c t r o n - n e u t r a l a n d e l e c t r o n ion c o l l i s i o n f r e q u e n c i e s . N e g l e c t ing s p a t i a l v a r i a t i o n s of t h e C o u l o m b l o g a r i t h m , Uei i s p r o p o r t i o n a l to t h e l o c a l e l e c t r o n d e n s i t y [3] w h e r e a s Ueo i s s p a c e i n d e p e n d e n t . The conductivity ratio thus becomes ~r/~i =
Ueo/W + ( n 2 / (~e )2) Uei / W
(1) 125
Volume 38A, number 2
PttYSICS
w h e r e t, ei is g i v e n by S p i t z e r ' s f o r m u l a u s i n g t h e a v e r a g e e l e c t r o n d e n s i t y . In o u r p r e s e n t e x p e r i m e n t a l c o n d i t i o n s c h a r g e l o s s e s a r e m a i n l y due to r e c o m b i n a t i o n , n e v e r t h e l e s s d i f f u s i o n l o s s e s are not completely negligible, the time constant i s of t h e o r d e r of o n e m i l l i s e c o n d . T h i s i n s u r e s that the radial distribution function is very close to a z e r o - o r d e r B e s s e l f u n c t i o n , w h e r e v e r t h e i n i t i a l d i s t r i b u t i o n m a y b e [11, 12]. T h i s y i e l d s 1.43 f o r t h e r a t i o n~e/(~e )2, a n d eq. (1) c a n b e written co ~ r / J i - ~eo = 1.43 ~ e i '
(2)
In fig. 1, t h e p o i n t s a r e m e a s u r e d v a l u e s of co ~ r / ~ i - ~eo v e r s u s e l e c t r o n d e n s i t y a n d t h e s o l i d l i n e i s t h e t h e o r e t i c a l v a l u e of Uei [2] c o r r e c t e d by t h e f a c t o r 1.43. T h e v a l u e of ~'eo h a s b e e n m e a s u r e d [131 u n d e r t h e s a m e e x p e r i m e n t a l c o n d i t i o n s b u t l a t e r in t h e a f t e r g l o w to e n s u r e low e n o u g h e l e c t r o n d e n s i t i e s a n d ~ei << ~eo" A t r o o m t e m p e r a t u r e it i s in good a g r e e m e n t w i t h p u b l i s h e d r e s u l t s [14]. T h e r e s u l t s of t h i s w o r k a r e in good a g r e e m e n t w i t h c l a s s i c a l t h e o r i e s of t h e e l e c t r o n ion i n t e r a c t i o n in p l a s m a s .
126
LETTERS
17 3anuary 1972
References [1] S . J . T a u b , J . - C . G a u t h i e r and J . - F . I ) e l p e c h , Prec. Tenth Intern. Conf. on Phenomena in ionized gases (Oxford, 1971) p. 259. [2] S . J . T a u b and J . - C . G a u t h i e r , Internal Report no. r i Itc c i r . [3} L. Spitzer, Physics of fully ionized gases (Interscience P u b l i s h e r s , New York, 1956). [.t] A.A. Dougal and L. Goldstein, Phys. Rev. 109 (1958) (315. [5] C.L. Chen, Phys. Rev. 135 (1964) A627. [6] C.R. Hoffmann and H.M, Skarsgard, Phys. Hey. 178 (1969) 168. [7] J. BouLmer and C. Vauge, C.R. Aead. Sci. 270 (1970) 573. [8] J.-C. Gauthier and J.-F. DeLpeeh, Rex'. Sci. Inst. 42 (1971) 958. [9] C. Sol and J. BouDner, C.R. Aeado Sci. 272 (1971) 727. [I01 L. Goldstein, Advances in electronics and electron physics, Vol. VII (Academic Press, New York, 1955) p. 399. [11] E.P.Gray and D.E.Kerr, Ann. Phys. 17 (1959) 27{i. [12] L. FrommhoLd and M. Biondi, Ann. Phys. 48 (1968) 407. I13] C. SoL, Internal Report no. 72/EG/1. [14] L.S. Frost and A.V. PheLps, Phys. Rev. 136 (1964) 1538.
ELECTRON-ION
PHYSICS
LETTERS
INTERACTION
IN
17 J a n u a r y 1972
A WEAKLY
IONIZED
GAS
*
J . - C . G A U T H I E R a n d C. SOL Institut d'Electronique Fondarnentale, Laboratoire associ~ au C. N. R. S., Universitb Paris-XI, 91-Orsay, France Received 22 November 1971
The effective e l e c t r o n - i o n collision frequency has been m e a s u r e d in a weakly ionized gas. Results a r e in good a g r e e m e n t with c l a s s i c a l t h e o r i e s for the e l e c t r o n - i o n interaction in p l a s m a s .
Recent computations [1,2] show that the high frequency conductivity ratio including electrone l e c t r o n i n t e r a c t i o n s c a n b e d i r e c t l y u s e d to d e t e r m i n e the e l e c t r o n ion c o l l i s i o n f r e q u e n c y if t h e r a t i o of t h e e l e c t r o n n e u t r a l to e l e c t r o n ion c o l l i s i o n f r e q u e n c y i s g r e a t e r t h a n 10 -1. O u r t h e o r y a n d S p i t z e r ' s c l a s s i c a l t h e o r y [3] a r e in g o o d a g r e e m e n t , in t h e i r c o m m o n r a n g e of v a lidity (weakly interacting plasmas), although they show a fairly large discrepancy with older e x p e r i m e n t a l r e s u l t s [4, 5]. H o f f m a n n a n d S k a r s g a r d h a v e p u b l i s h e d r e s u l t s [6], u s i n g c a v ity t e c h n i q u e s , w h i c h s e e m to i n d i c a t e good agreement with Spitzer's theory over a limited r a n g e of e l e c t r o n d e n s i t i e s ((1 - 5) x 1010 c m - 3 ) . W e r e p o r t h e r e m i c r o w a v e m e a s u r e m e n t s of t h e electron-ion collision frequency for momentum t r a n s f e r in a r o o m t e m p e r a t u r e a r g o n a f t e r g l o w a t a p r e s s u r e of 5 t o r r w h i c h a r e in good a g r e e m e n t w i t h o u r t h e o r y in t h e r a n g e 4 x 109 8 x 1 0 1 0 c m - 3 . M e a s u r e m e n t s w e r e m a d e 800 to 1200 ~ s a f t e r t h e e n d of t h e a c t i v e d i s c h a r g e . The plasma is contained inside a cylindrical g l a s s b o t t l e of 40 c m l e n g t h a n d 9 m m i n s i d e d i a m e t e r . It i s i n s e r t e d a l o n g t h e a x i s of a standard X-band waveguide; our microwave diagnostic system has been described elsewhere [7-9]. We take into account the residual microw a v e r e f l e c t i o n s a n d l o s s e s of t h e g a s b o t t l e . T h e e l e c t r o n g a s i s in e q u i l i b r i u m w i t h the n e u t r a l s a t r o o m t e m p e r a t u r e (295°K) w i t h i n 10°K, and the high electron thermal conductivity insures a spatially uniform electron temperat u r e [1]. It i s w e l l k n o w n t h a t , in t h e h i g h f r e q u e n c y l i m i t u 2 / w 2 << 1, t h e c o n d u c t i v i t y r a t i o b e t w e e n t h e r e a l p a r t (~r a n d t h e i m a g i n a r y p a r t * This work was supported by D. R. M. E. under cont r a c t no. 70/670.
F
"~ - - ~ - - - ~ ]
]
]
I
I
I --
3
lO 9 _
10 8
• 1010
__ t _ _ ± ~ n e (cm -3)
1011
Fig. 1. Plot of experimental and theoretical electron ion collision frequency at room t e m p e r a t u r e , v e r s u s average density, ~i of t h e h i g h f r e q u e n c y c o m p l e x c o n d u c t i v i t y [10] b e c o m e s ~ r / ~ i = Ueffne/O)ne, w h e r e t h e b a r d e n o t e s s p a t i a l a v e r a g i n g . In a w e a k l y i o n i z e d g a s , it h a s b e e n s h o w n [2] t h a t t h e t o t a l c o l l i s i o n f r e q u e n c y Ueff c a n b e w r i t t e n Ueff = Uei + Veo w h e r e Ueo a n d Uei a r e r e s p e c t i v e l y t h e e l e c t r o n - n e u t r a l a n d e l e c t r o n ion c o l l i s i o n f r e q u e n c i e s . N e g l e c t ing s p a t i a l v a r i a t i o n s of t h e C o u l o m b l o g a r i t h m , Uei i s p r o p o r t i o n a l to t h e l o c a l e l e c t r o n d e n s i t y [3] w h e r e a s Ueo i s s p a c e i n d e p e n d e n t . The conductivity ratio thus becomes ~r/~i =
Ueo/W + ( n 2 / (~e )2) Uei / W
(1) 125
Volume 38A, number 2
PttYSICS
w h e r e t, ei is g i v e n by S p i t z e r ' s f o r m u l a u s i n g t h e a v e r a g e e l e c t r o n d e n s i t y . In o u r p r e s e n t e x p e r i m e n t a l c o n d i t i o n s c h a r g e l o s s e s a r e m a i n l y due to r e c o m b i n a t i o n , n e v e r t h e l e s s d i f f u s i o n l o s s e s are not completely negligible, the time constant i s of t h e o r d e r of o n e m i l l i s e c o n d . T h i s i n s u r e s that the radial distribution function is very close to a z e r o - o r d e r B e s s e l f u n c t i o n , w h e r e v e r t h e i n i t i a l d i s t r i b u t i o n m a y b e [11, 12]. T h i s y i e l d s 1.43 f o r t h e r a t i o n~e/(~e )2, a n d eq. (1) c a n b e written co ~ r / J i - ~eo = 1.43 ~ e i '
(2)
In fig. 1, t h e p o i n t s a r e m e a s u r e d v a l u e s of co ~ r / ~ i - ~eo v e r s u s e l e c t r o n d e n s i t y a n d t h e s o l i d l i n e i s t h e t h e o r e t i c a l v a l u e of Uei [2] c o r r e c t e d by t h e f a c t o r 1.43. T h e v a l u e of ~'eo h a s b e e n m e a s u r e d [131 u n d e r t h e s a m e e x p e r i m e n t a l c o n d i t i o n s b u t l a t e r in t h e a f t e r g l o w to e n s u r e low e n o u g h e l e c t r o n d e n s i t i e s a n d ~ei << ~eo" A t r o o m t e m p e r a t u r e it i s in good a g r e e m e n t w i t h p u b l i s h e d r e s u l t s [14]. T h e r e s u l t s of t h i s w o r k a r e in good a g r e e m e n t w i t h c l a s s i c a l t h e o r i e s of t h e e l e c t r o n ion i n t e r a c t i o n in p l a s m a s .
126
LETTERS
17 3anuary 1972
References [1] S . J . T a u b , J . - C . G a u t h i e r and J . - F . I ) e l p e c h , Prec. Tenth Intern. Conf. on Phenomena in ionized gases (Oxford, 1971) p. 259. [2] S . J . T a u b and J . - C . G a u t h i e r , Internal Report no. r i Itc c i r . [3} L. Spitzer, Physics of fully ionized gases (Interscience P u b l i s h e r s , New York, 1956). [.t] A.A. Dougal and L. Goldstein, Phys. Rev. 109 (1958) (315. [5] C.L. Chen, Phys. Rev. 135 (1964) A627. [6] C.R. Hoffmann and H.M, Skarsgard, Phys. Hey. 178 (1969) 168. [7] J. BouLmer and C. Vauge, C.R. Aead. Sci. 270 (1970) 573. [8] J.-C. Gauthier and J.-F. DeLpeeh, Rex'. Sci. Inst. 42 (1971) 958. [9] C. Sol and J. BouDner, C.R. Aeado Sci. 272 (1971) 727. [I01 L. Goldstein, Advances in electronics and electron physics, Vol. VII (Academic Press, New York, 1955) p. 399. [11] E.P.Gray and D.E.Kerr, Ann. Phys. 17 (1959) 27{i. [12] L. FrommhoLd and M. Biondi, Ann. Phys. 48 (1968) 407. I13] C. SoL, Internal Report no. 72/EG/1. [14] L.S. Frost and A.V. PheLps, Phys. Rev. 136 (1964) 1538.