Electron-neutral heat transfer in plasmas

Volume 25A. n u m b e r 7

PHYSICS

ELECTRON-NEUTRAL

HEAT

LETTERS

9 O c t o b e r 1967

TRANSFER

IN

PLASMAS*

K. J . N Y G A A R D S p e r r y R a n d R e s e a r c h C e n t e r . S u d b u r y . M a s s a c h u s e t / s 01776

Received 14 Augaast 1967

Pulsed heat-flow e x p e r i m e n t s in afterglows of helium and xenon have yielded data on the electron t h e r m a l diffusivity. E l e c t r o n - t e m p e r a t u r e t r a n s i e n t s were m e a s u r e d by taking advantage of the t e m p e r a t u r e s e n s i tivity of e l e c t r o - a c o u s t i c (Tonks-Dattner) resonances. The r e s u l t s agree with Shkarofsky's theory in the l i m i t of p r e d o m i n a n t e l e c t r o n - n e u t r a l collisions.

In t h i s c o m m u n i c a t i o n we c o n s i d e r h e a t t r a n s f e r in a w e a k l y i o n i z e d p l a s m a w i t h a l l c o m p o nents at the same initial equilibrium temperature. If t h e e l e c t r o n s in t h i s p l a s m a a r e s l i g h t l y h e a t e d by a b s o r p t i o n of a p u l s e of m i c r o w a v e r a d i a t i o n , fundamental transport quantities can de determ i n e d by o b s e r v i n g t h e s u b s e q u e n t h e a t flow. F o r o u r p u r p o s e it i s c o n v e n i e n t t o i n t r o d u c e t h e electron thermal diffusivity DT, which is defined

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D w = cK(~ k n e ) -1 ,

(1)

where ~ is the conventional thermal conductivity for the electrons, k is Boltzmann's constant, and n e is the electron number density. The objective of o u r w o r k h a s b e e n to m e a s u r e D T i n r o o m temperature afterglowplasmas with electron dens i t i e s r a n g i n g f r o m 1 0 g t o 1011 c m -3. R e s u l t s in n e o n , a n d a d e t a i l e d d e s c r i p t i o n of t h e m e t h o d , h a v e a l r e a d y b e e n p u b l i s h e d [1]. H e r e we p r e s e n t additional data for helium and xenon. F o l l o w i n g s t a n d a r d n o m e n c l a t u r e , we s h a l l call the plasma weakly ionized when the electronn e u t r a l c o l l i s i o n f r e q u e n c y Uen i s m u c h l a r g e r t h a n t h e e l e c t r o n - i o n c o l l i s i o n f r e q u e n c y Uei. In this limit the electron thermal diffusivity is given

by [2] 10kTeo D T-

3rag Kuen

,

fo¢r

(2)

where Teo and m are the electron temperature and mass, respectively, and the coefficient gK is s e n s i t i v e to t h e v e l o c i t y d e p e n d e n c e of eenT h e p r i n c i p I e of t h e e x p e r i m e n t i s t o h e a t a s m a l l v o l u m e of t h e e l e c t r o n s in a n a f t e r g l o w * Supported in p a r t by Air F o r c e Cambridge R e s e a r c h L a b o r a t o r i e s u n d e r C o n t r a c t No. F1962867C0096.

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<°.)c=~ 3) Fig. 1. Electron t h e r m a l diffusivity in helium as a function of average electron density. The vertical b a r s i n dicate maximum s c a t t e r in the e x p e r i m e n t . The arrow at ~ 4.5 x 1010 c m - 3 shows the density at which Yen ~ ~ei. Spitzer and H~lrm's c u r v e was calculated for Teo = 300OK. plasma with a short pulse (effectively a 5-impulse} of m i c r o - w a v e r a d i a t i o n , w h i c h i s a p p l i e d a t t h e p o s i t i o n z= 0 a t t i m e t = 0. F r o m s i m p l e o n e d i m e n s i o n a l h e a t - t r a n s f e r c o n s i d e r a t i o n s [1,3] we f i n d t h a t t h e t e m p e r a t u r e i n c r e a s e at a p o s i tion z is ATe(z,t)

- Te(z,t

) _ T e o cc

o~ { e x p [ _ z 2 / ( 4 D T t

) _ l/ren]}/(l

" (3)

w h e r e Ten i s t h e e f f e c t i v e r e l a x a t i o n t i m e f o r c o l lisional energy transfer from electrons to neut r a l s a t o m s . U n f o r t u n a t e l y , in m a n y c a s e s Ten i s not accurately known. Nevertheless, D r can be d e t e r m i n e d f r o m t h e l o g a r i t h m i c r a t i o of e l e c t r o n . 567

Volume 25A. number 7 \

PHYSICS LETTERS

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SPITZER ond H A R M N XENON

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Fig. 2. Electron thermal diffusivity in xenon. The points from Goldstein and Sekiguchi were obtained in their steady state method, which is supposed to be more accurate than their transient method. t e m p e r a t u r e t r a n s i e n t s m e a s u r e d at two d i f f e r e n t positions, z 1 and z2, but at the s a m e t i m e / r , i . e .,

D w = (z 2 -z21)/t 4t r In [ ~ T e ( Z l , t r ) / A T e ( Z 2 , tr)]}. (4) Notice that ten is not included in eq. (4) and that only r e l a t i v e t e m p e r a t u r e i n c r e a s e s a r e needed. The t e m p e r a t u r e r a t i o in eq. (4) is d e t e r m i n e d by m e a n s of T o n k s - D a t t n e r r e s o n a n c e s e x c i t e d by a s t r i p l i n e . The p r o p e r t i e s of such r e s o n a n c e s in a f t e r g l o w s have been d i s c u s s e d by S c h m it t [4] and t h e i r application as an a c o u s t i c d e t e c t o r [5] and e l e c t r o n " t h e r m o m e t e r " [1] has r e c e n t l y been developed. The m e a s u r e m e n t s in h e l i u m (fig. 1) w e r e made at 5 0 0 # s e c and l a t e r a f t e r breakdown. E x p e r i m e n t s of I n g r a h a m and Brown [6] f o r s i m i l a r conditions show that e q u a l i z a t i o n of e l e c t r o n and atom t e m p e r a t u r e s is e s t a b l i s h e d by this t i m e . We notice that at v e r y low e l e c t r o n d e n s i t i e s the m e a s u r e d v a l u e s of DT tend to be constant, as e x p e c t e d f r o m eq. (2). The h o r i z o n t a l a r r o w at D T = 128 m 2 / s e c g i v e s the l im i t as U e i / V e n - ~ 0 and was c a l c u l a t e d f r o m S h k a r o f s k y ' s t h e o r y [2] using P a c k and P h e l p s ' value f o r the m o m e n t u m t r a n s f e r c r o s s s ect i o n [7] (~= 5.3 × 10 -16 cm2). In h e l i u m , Uen is p r o p o r t i o n a l to the e l e c t r o n v e l o c i t y , and f o r this p a r t i c u l a r c a s e the c o e f f i cient in eq. (2) is gK = 1.47. As the e l e c t r o n density (and hence the e l e c t r o n - i o n c o l l i s i o n f r e q u e n cy) i n c r e a s e s , the t h e r m a l diffusivity tends to d e c r e a s e . The p r e s e n t method u t il iz in g T o n k s D a t t n e r r e s o n a n c e s can only be used when the e l e c t r o n density p r o f i l e is known. In our e x p e r i m e n t this l i m i t s its a p p l i c a b i l i t y to a m b i p o l a r dif568

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9 October 1967

f u s i o n - c o n t r o l l e d p l a s m a (densities between 109 and 1011 c m - 3 ) . F o r t u n a t e l y , Sekiguchi and H e r n d o n [ 8 ] have made m e a s u r e m e n t s c l o s e to 1012 c m - 3 by taking advantage of the quenching of r e c o m b i n a t i o n r ad i at i o n [9]. F o r c o m p a r i s o n , in fig. 1 we have plotted S p i t z e r and H~irm's c a l culation [10] f o r an e f f e c t i v e l y fully ionized p l a s ma. This c u r v e s e e m s to bridge the gap between Sekiguchi and H e r n d o n ' s [8] and our data. A compilation of data in xenon is d i sp l ay ed in fig. 2. The open c i r c l e s w e r e r e p o r t e d by Goldstein and Sekiguchi [3] and obtained by m e a n s of the r e c o m b i n a t i o n quenching method. The f i l l e d c i r c l e s r e p r e s e n t m e a s u r e m e n t s em p l o y i n g Tonks D at t n er r e s o n a n c e s for d e t e r m i n a t i o n of e l e c t r o n t e m p e r a t u r e i n c r e a s e s . Our m e a s u r e m e n t s w e r e m a d e m o r e than 4 m s e c a f t e r breakdown, and an e l e c t r o n t e m p e r a t u r e of 300°K was a s s u m e d . T h e o r e t i c a l e s t i m a t e s of the t h e r m a l diffusivity w e r e obtained by using Ph el p s, Fundingsland and B r o w n ' s value f o r the m o m e n t u m t r a n s f e r c r o s s sect i o n [111 (5.1 × 1015 cm2). The upper value at D T = 109 m Z / s e c was c a l c u l a t e d with g K = 0.589, and the l o w e r a r r o w at D T = 53 m 2 / s e c with g K = = 0.286. T h e s e v a l u e s of g K c o r r e s p o n d to c r o s s s e c t i o n s p r o p o r t i o n a l to v - 3 and u-2, r e s p e c t i v e l y Such v e l o c i t y d e p e n d e n c i e s have been c o n s i d e r e d by Pack, V o sh al l and P h e l p s [12]. Two of the high density points of Goldstein and Sekiguchi [3] fall c l e a r l y above the expected l i m i t f o r the fully ionized p l a s m a in which D T cc TeZ . The explanation f o r this d i s c r e p a n c y is probably that t h e s e o b s e r v a t i o n s w e r e made e a r l y in the a f t e r g l o w (200 - 4 0 0 sec a f t e r breakdown), when the e l e c t r o n t e m p e r a t u r e may have been above e q u i l i b r i u m t e m p e r a t u r e of 300°K as s u g g e s t e d by Sekiguchi and Herndon [8]. The author is indebted to J. E. M o r r i s for a s s i s t a n c e in the l a b o r a t o r y and to C . D . Lustig and P. M. Stone for c o m m e n t s on the m a n u s c r i p t . 1. K.J. Nygaard, Phys. Rev. 157 (1967) 138. 2. I.P. Shkarofsky, Can. J. Phys. 39 (1961) 1619. 3. L. Goldstein and T.Sekiguchi, Phys. Rev. 109 (1958) 625. 4. H.J. Schmitt, Appl. Phys. Letters 6 (1965) 187. 5. K.J. Nygaard, Phys. Letters 20 (1966) 370. 6. J. C. Ingraham and S. C. Brown, Phys. Rev. 138 {1965) A 1015. 7. J. L. Pack and A. V. Phelps, Phys. Rev. 121 (1961) 798. 8. T. Sekiguchi and R. C. Herndon, Phys. Rev. 112 (1958) 1. 9. L. Goldstein, J. M. Anderson and G. L. Clark, Phys. Rev. 90 (1953) 486. 10. L. Spitzer Jr. and R.H~irm, Phys. Rev. 89 (1953) 977. 11. A.V. Phelps, O.T. Fundingsland and S. C. Brown, Phys. Rev. 84 (1951) 559. 12. J. L. Pack, R. E. Voshall and A. V. Phelps, Phys. Rev. 127 (1962) 2084.