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Electron heating by beam-plasma interaction in uniform magnetic field
Volume28A, number 5
ELECTRON
PHYSICS
HEATING BY IN UNIFORM
LETTERS
16 December 1968
BEAM-PLASMA INTERACTION MAGNETIC FIELD
J. JANCAI~IK, V. P I F F L and M. S E I D L Institute of Plasma Physics, Czechoslovak Academy of Sciences, Prague, Czechoslovakia Received 12 August 1968
Experimental evidence is given for the enhancement of r.f. electron heating by cyclotron resonance absorption in a cylindrical layer where the frequency of the r.f. field excited by electron beam-plasma interaction is equal to the local hybrid frequency.
H i g h - f r e q u e n c y f i e l d s r e s u l t i n g f r o m the i n t e r a c t i o n of an e l e c t r o n b e a m w i t h p l a s m a in a u n i f o r m m a g n e t i c f i e l d s t r o n g l y a f f e c t the d i s t r i b u t i o n f u n c t i o n s of both the b e a m and p l a s m a e l e c t r o n s . T h e l a t t e r e f f e c t m a n i f e s t s i t s e l f by the h e a t i n g of the o r i g i n a l l y c o l d p l a s m a . T h e s i m plest evidence for plasma heating is r.f. breakdown of the b e a m - p l a s m a d i s c h a r g e w h i c h o c c u r s w h e n the p l a s m a e l e c t r o n s c o n s t i t u t e the m a i n i o n i z i n g a g e n t [ 1- 5]. T h e h e a t i n g i s due to t h e f o l l o w i n g two t y p e s of w a v e s w h i c h c a n be e x c i t e d by the e l e c t r o n b e a m in a h o m o g e n e o u s c o l d p l a s m a in a u n i f o r m m a g n e t i c field: the l o w - f r e q u e n c y w a v e , w < ¢Oce, r e f e r r e d to a s t h e w a v e , a n d t h e h i g h - f r e q u e n c y w a v e , co > Wee , r e f e r r e d to a s t h e L w a v e [6]. In a l l e x p e r i m e n t s t h e s e w a v e s a p p e a r in s h o r t p u l s e s of about 0.1 ~s d u r a t i o n [6-8] s o that t h e y are well suited for stochastic heating. The latter c o u l d b e s i g n i f i c a n t l y e n h a n c e d by c y c l o t r o n r e s o n a n c e w h i c h i s p o s s i b l e f o r the H w a v e s e x c i t e d n e a r to c y c l o t r o n h a r m o n i c s [9]. T h e d i f f i c u l t y with s u c h a m o d e l in a u n i f o r m p l a s m a l i e s in the f a c t that no w a v e s a r e e x c i t e d if the c y c l o t r o n d a m p i n g i s l a r g e [6]. T h e s i t u a t i o n b e c o m e s d i f f e r e n t if one t a k e s into a c c o u n t t h a t the p l a s m a d e n s i t y d e c r e a s e s w i t h r a d i u s . S i n c e c y c l o t r o n d a m p i n g n e a r the q - t h h a r m o n i c s i s p r o p o r t i o n a l to (k±p)2(q- 1) e x p - { ( c o - q c o c e ) / k z vt} 2 . p b e i n g the e l e c t r o n L a r m o r r a d i u s , w a v e s with k i p << 1 a r e only s l i g h t l y d a m p e d . A s the w a v e p r o p a g a t e s a c r o s s the m a g n e t i c f i e l d into a r e g i o n of l o w e r p l a s m a d e n s i t y , k j_ i n c r e a s e s until it r e a c h e s a high v a l u e n e a r the h y b r i d r e s o n a n c e co2 + w~2 H e r e k±p ~- 1, and c y c l o t r o n d a m p . = ¢o2 15 t;" mg becomes large. As a result, waves are ex-
c i t e d in the d e n s e p l a s m a c o r e and a b s o r b e d in a c y l i n d r i c a l l a y e r w h e r e the e x c i t e d f r e q u e n c y i s e q u a l to the l o c a l h y b r i d f r e q u e n c y . T h i s a b s o r p tion m e c h a n i s m h a s b e e n p r o p o s e d in r e f . 10 and r e c e n t l y d i s c u s s e d in r e f . 11. We p r e s e n t e x p e r i m e n t a l e v i d e n c e of the e x i s t e n c e of the a b o v e e f fect. A p a r a l l e l flow h o l l o w e l e c t r o n b e a m (5 kV, 0.2 A) w a s i n j e c t e d a l o n g a u n i f o r m m a g n e t i c f i e l d of 1000 G. T h e b e a m p r o d u c e d a p l a s m a c o l u m 50 c m l o n g by i o n i z i n g a r g o n g a s a d m i t t e d at a p r e s s u r e 1.5 × 10 - 4 T o r r . T h e p l a s m a (n = = 1012 c m - 3 ) w a s s u s t a i n e d by the H w a v e e x c i t ed n e a r to the 3 r d c y c l o t r o n h a r m o n i c s (the L w a v e w a s a b s e n t at t h i s d e n s i t y ) . By m e a n s of an e l e c t r o s t a t i c m u l t i g r i d a n a l y s e r p l a c e d at the c o l l e c t o r end the d i s t r i b u t i o n in p a r a l l e l v e l o c i t i e s of the e l e c t r o n s l e a v i n g the p l a s m a w a s m e a s u r ed. T h e e n t r a n c e h o l e of the a n a l y s e r w a s m o v a b l e in the r a d i a l d i r e c t i o n . The e n e r g y d i s t r i b u tion of the e l e c t r o n s c o n s i s t e d of two w e l l s e p a r a t e d p a r t s [11]. T h e low e n e r g y p a r t , e x t e n d i n g to about 400 eV, w a s m a d e up the h e a t e d p l a s m a e l e c t r o n s . M e a s u r e m e n t of the e n e r g y d i s t r i b u tion at v a r i o u s r a d i i showed that the e l e c t r o n " t e m p e r a t u r e " h a s a s h a r p m a x i m u m at a r a d i u s r = 16 ram. At t h i s r a d i u s the a m p l i t u d e of the r . f . f i e l d v a n i s h e d i n d i c a t i n g the o c c u r e n c e of h y b r i d r e s o n a n c e . T h i s i s shown in fig. 1 w h e r e the r a t i o (I100 - I400)/(IO - /400) i s e q u a l to the p e r c e n t a g e of the f a s t e l e c t r o n s c o n t a i n e d in the t a i l of the low e n e r g y h u m p of the d i s t r i b u t i o n . It m a y b e n o t e d that the ion c u r r e n t ( p r o p o r t i o n a l to p l a s m a d e n s i t y ) d e c r e a s e s s l o w l y with r a d i u s in c o n t r a s t to the e l e c t r o n c u r r e n t w h i c h i s high but in the b e a m r e g i o n w h e r e the r . f . f i e l d i s s t r o n g . T h i s i n d i c a t e s that only the f a s t e l e c t r o n s a r e l e a v i n g the p l a s m a w h i l e the c o l d e l e c t r o n s 331
PHYSICS
Volume 28A, number 5
1 .
A \
16 D e c e m b e r 1968
LETTERS
60 V a s m e a s u r e d b y t h e e n e r g y of t h e i o n s . The above measurements indicate that although c y c l o t r o n r e s o n a n c e e n h a n c e s t h e h e a t i n g of a s m a l l p a r t of p l a s m a e l e c t r o n s m o s t e l e c t r o n s a r e h e a t e d by n o n r e s o n a n t s t o c h a s t i c h e a t i n g in regions where the r.f. field has a high amplitude.
,:l Reference s
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Fig. 1. Radial dependence of e l e c t r o n c u r r e n t s m e a s ured by the analyser. Ix is the c u r r e n t density made up of electrons leaving~tfie p l a s m a with an energy g r e a t e r than x elect~ou~ volts, so t h a t / 4 0 0 is proportional to the beam c u r r e n t density while (I0 - •400) c o r r e s p o n d s to the heated p l a s m a e l e c t r o n s . Ion c u r r e n t and r.f. field amplitude are also indicated. a r e e l e c t r o s t a t i c a l l y confined by the s h e a t h effect. T h e p o t e n t i a l d i f f e r e n c e on t h e s h e a t h w a s a b o u t
SUPERLARGE-SCALE
MODEL
1. w . D. Getty and L. D. Smullin, J. Appl. Phys. 34 (1963} 3421. 2. E . A . Kornilov, O. F. Kovpik, Ya. B. Fainberg and I. F. Kharchenko, Fizika P l a s m y 4 (Kiev 1965) p. 145. 3. H.J. Hopman and T. Matitti, 8th Int. Conf. on P h e nomena in ionized g a s e s , Vienna 1967, p. 384. 4. V. Piffl, M. Seidl, P. ~unka and J . Ullschmied, Third Czech. Conf. on Electronics, Prague, 1965, p. 51. 5. A.K. Berezin, Ya. B. Fainberg, L. I. Bolotin and G. P. Berezina, A t . Energiya 18 (1965) 5. 6. M. Seidl and P. Sunka, Nuclear Fusion 7 (1967) 237. 7. A.K. Berezin, Ya. B. F a i n b e r g and I. A. Bezyazychnyi, Zh. Eksp. i Teor. Fiz. L e t t e r s 7 (1968) 156. 8. H . J . Hopman and W. Ott, P l a s m a Physics 10 (1968) 315. 9. T.H. Stix, Phys. Fluids 7 (1964) 1960. 10. T.H. Stix, Phys. Rev. L e t t e r s 15 (1965) 878. 11. J. Jan6af-lk et al., Generation and heating of p l a s m a by b e a m - p l a s m a interaction, P r o c . Third Conf. on P l a s m a p h y s i c s and controlled nuclear fusion Res. Novosibirisk, 1968, to be published.
OF
EINSTEIN
UNIVERSE
Z. H O t ~ K Department of Physics, Technical University, Praha, Czechoslovakia Received 12 August 1968
A closed relativistic s t e a d y - s t a t e model of the universe consisting of a very large number of oscillating "local u n i v e r s e s " is proposed. It c r e a t e s a nearly constant s c a l a r gravitational potential with average value - c 2, is in full a g r e e m e n t with Mach's principle and is not c o n t r a r y Lo astronomical observations.
T h e E i n s t e i n s t a t i c u n i v e r s e s e e m s not to b e a c c e p t a b l e a t p r e s e n t b e c a u s e of i t s i n s t a b i l i t y and disagreement with the observed red shift. M o r e o v e r , r e c e n t o b s e r v a t i o n s showed that the u n i v e r s e w a s v e r y m u c h d e n s e r in t h e p a s t t h a n it i s a t p r e s e n t . T h i s m e a n s t h a t t h e o b s e r v a b l e 332
p a r t of t h e u n i v e r s e d o e s not s a t i s f y t h e p e r f e c t cosmological principle. On t h e o t h e r h a n d , H o y l e a n d N a r l i k a r i n v e s t i g a t e d in 1965 [1] t h e c a s e in w h i c h t h e r e a r e d e p a r t u r e s f r o m h o m o g e n e i t y in t h e u n i v e r s e . T h e y found that t h e r e will be m a n y l o c a l i z e d finite o s -
ELECTRON
PHYSICS
HEATING BY IN UNIFORM
LETTERS
16 December 1968
BEAM-PLASMA INTERACTION MAGNETIC FIELD
J. JANCAI~IK, V. P I F F L and M. S E I D L Institute of Plasma Physics, Czechoslovak Academy of Sciences, Prague, Czechoslovakia Received 12 August 1968
Experimental evidence is given for the enhancement of r.f. electron heating by cyclotron resonance absorption in a cylindrical layer where the frequency of the r.f. field excited by electron beam-plasma interaction is equal to the local hybrid frequency.
H i g h - f r e q u e n c y f i e l d s r e s u l t i n g f r o m the i n t e r a c t i o n of an e l e c t r o n b e a m w i t h p l a s m a in a u n i f o r m m a g n e t i c f i e l d s t r o n g l y a f f e c t the d i s t r i b u t i o n f u n c t i o n s of both the b e a m and p l a s m a e l e c t r o n s . T h e l a t t e r e f f e c t m a n i f e s t s i t s e l f by the h e a t i n g of the o r i g i n a l l y c o l d p l a s m a . T h e s i m plest evidence for plasma heating is r.f. breakdown of the b e a m - p l a s m a d i s c h a r g e w h i c h o c c u r s w h e n the p l a s m a e l e c t r o n s c o n s t i t u t e the m a i n i o n i z i n g a g e n t [ 1- 5]. T h e h e a t i n g i s due to t h e f o l l o w i n g two t y p e s of w a v e s w h i c h c a n be e x c i t e d by the e l e c t r o n b e a m in a h o m o g e n e o u s c o l d p l a s m a in a u n i f o r m m a g n e t i c field: the l o w - f r e q u e n c y w a v e , w < ¢Oce, r e f e r r e d to a s t h e w a v e , a n d t h e h i g h - f r e q u e n c y w a v e , co > Wee , r e f e r r e d to a s t h e L w a v e [6]. In a l l e x p e r i m e n t s t h e s e w a v e s a p p e a r in s h o r t p u l s e s of about 0.1 ~s d u r a t i o n [6-8] s o that t h e y are well suited for stochastic heating. The latter c o u l d b e s i g n i f i c a n t l y e n h a n c e d by c y c l o t r o n r e s o n a n c e w h i c h i s p o s s i b l e f o r the H w a v e s e x c i t e d n e a r to c y c l o t r o n h a r m o n i c s [9]. T h e d i f f i c u l t y with s u c h a m o d e l in a u n i f o r m p l a s m a l i e s in the f a c t that no w a v e s a r e e x c i t e d if the c y c l o t r o n d a m p i n g i s l a r g e [6]. T h e s i t u a t i o n b e c o m e s d i f f e r e n t if one t a k e s into a c c o u n t t h a t the p l a s m a d e n s i t y d e c r e a s e s w i t h r a d i u s . S i n c e c y c l o t r o n d a m p i n g n e a r the q - t h h a r m o n i c s i s p r o p o r t i o n a l to (k±p)2(q- 1) e x p - { ( c o - q c o c e ) / k z vt} 2 . p b e i n g the e l e c t r o n L a r m o r r a d i u s , w a v e s with k i p << 1 a r e only s l i g h t l y d a m p e d . A s the w a v e p r o p a g a t e s a c r o s s the m a g n e t i c f i e l d into a r e g i o n of l o w e r p l a s m a d e n s i t y , k j_ i n c r e a s e s until it r e a c h e s a high v a l u e n e a r the h y b r i d r e s o n a n c e co2 + w~2 H e r e k±p ~- 1, and c y c l o t r o n d a m p . = ¢o2 15 t;" mg becomes large. As a result, waves are ex-
c i t e d in the d e n s e p l a s m a c o r e and a b s o r b e d in a c y l i n d r i c a l l a y e r w h e r e the e x c i t e d f r e q u e n c y i s e q u a l to the l o c a l h y b r i d f r e q u e n c y . T h i s a b s o r p tion m e c h a n i s m h a s b e e n p r o p o s e d in r e f . 10 and r e c e n t l y d i s c u s s e d in r e f . 11. We p r e s e n t e x p e r i m e n t a l e v i d e n c e of the e x i s t e n c e of the a b o v e e f fect. A p a r a l l e l flow h o l l o w e l e c t r o n b e a m (5 kV, 0.2 A) w a s i n j e c t e d a l o n g a u n i f o r m m a g n e t i c f i e l d of 1000 G. T h e b e a m p r o d u c e d a p l a s m a c o l u m 50 c m l o n g by i o n i z i n g a r g o n g a s a d m i t t e d at a p r e s s u r e 1.5 × 10 - 4 T o r r . T h e p l a s m a (n = = 1012 c m - 3 ) w a s s u s t a i n e d by the H w a v e e x c i t ed n e a r to the 3 r d c y c l o t r o n h a r m o n i c s (the L w a v e w a s a b s e n t at t h i s d e n s i t y ) . By m e a n s of an e l e c t r o s t a t i c m u l t i g r i d a n a l y s e r p l a c e d at the c o l l e c t o r end the d i s t r i b u t i o n in p a r a l l e l v e l o c i t i e s of the e l e c t r o n s l e a v i n g the p l a s m a w a s m e a s u r ed. T h e e n t r a n c e h o l e of the a n a l y s e r w a s m o v a b l e in the r a d i a l d i r e c t i o n . The e n e r g y d i s t r i b u tion of the e l e c t r o n s c o n s i s t e d of two w e l l s e p a r a t e d p a r t s [11]. T h e low e n e r g y p a r t , e x t e n d i n g to about 400 eV, w a s m a d e up the h e a t e d p l a s m a e l e c t r o n s . M e a s u r e m e n t of the e n e r g y d i s t r i b u tion at v a r i o u s r a d i i showed that the e l e c t r o n " t e m p e r a t u r e " h a s a s h a r p m a x i m u m at a r a d i u s r = 16 ram. At t h i s r a d i u s the a m p l i t u d e of the r . f . f i e l d v a n i s h e d i n d i c a t i n g the o c c u r e n c e of h y b r i d r e s o n a n c e . T h i s i s shown in fig. 1 w h e r e the r a t i o (I100 - I400)/(IO - /400) i s e q u a l to the p e r c e n t a g e of the f a s t e l e c t r o n s c o n t a i n e d in the t a i l of the low e n e r g y h u m p of the d i s t r i b u t i o n . It m a y b e n o t e d that the ion c u r r e n t ( p r o p o r t i o n a l to p l a s m a d e n s i t y ) d e c r e a s e s s l o w l y with r a d i u s in c o n t r a s t to the e l e c t r o n c u r r e n t w h i c h i s high but in the b e a m r e g i o n w h e r e the r . f . f i e l d i s s t r o n g . T h i s i n d i c a t e s that only the f a s t e l e c t r o n s a r e l e a v i n g the p l a s m a w h i l e the c o l d e l e c t r o n s 331
PHYSICS
Volume 28A, number 5
1 .
A \
16 D e c e m b e r 1968
LETTERS
60 V a s m e a s u r e d b y t h e e n e r g y of t h e i o n s . The above measurements indicate that although c y c l o t r o n r e s o n a n c e e n h a n c e s t h e h e a t i n g of a s m a l l p a r t of p l a s m a e l e c t r o n s m o s t e l e c t r o n s a r e h e a t e d by n o n r e s o n a n t s t o c h a s t i c h e a t i n g in regions where the r.f. field has a high amplitude.
,:l Reference s
I ,®!
o
II/i
/
i 6
"
'
'
4o
'
4a
"
4
,o
\ '
4
. nAOa~
"
4
'
~o
'
~2
o
le~l
Fig. 1. Radial dependence of e l e c t r o n c u r r e n t s m e a s ured by the analyser. Ix is the c u r r e n t density made up of electrons leaving~tfie p l a s m a with an energy g r e a t e r than x elect~ou~ volts, so t h a t / 4 0 0 is proportional to the beam c u r r e n t density while (I0 - •400) c o r r e s p o n d s to the heated p l a s m a e l e c t r o n s . Ion c u r r e n t and r.f. field amplitude are also indicated. a r e e l e c t r o s t a t i c a l l y confined by the s h e a t h effect. T h e p o t e n t i a l d i f f e r e n c e on t h e s h e a t h w a s a b o u t
SUPERLARGE-SCALE
MODEL
1. w . D. Getty and L. D. Smullin, J. Appl. Phys. 34 (1963} 3421. 2. E . A . Kornilov, O. F. Kovpik, Ya. B. Fainberg and I. F. Kharchenko, Fizika P l a s m y 4 (Kiev 1965) p. 145. 3. H.J. Hopman and T. Matitti, 8th Int. Conf. on P h e nomena in ionized g a s e s , Vienna 1967, p. 384. 4. V. Piffl, M. Seidl, P. ~unka and J . Ullschmied, Third Czech. Conf. on Electronics, Prague, 1965, p. 51. 5. A.K. Berezin, Ya. B. Fainberg, L. I. Bolotin and G. P. Berezina, A t . Energiya 18 (1965) 5. 6. M. Seidl and P. Sunka, Nuclear Fusion 7 (1967) 237. 7. A.K. Berezin, Ya. B. F a i n b e r g and I. A. Bezyazychnyi, Zh. Eksp. i Teor. Fiz. L e t t e r s 7 (1968) 156. 8. H . J . Hopman and W. Ott, P l a s m a Physics 10 (1968) 315. 9. T.H. Stix, Phys. Fluids 7 (1964) 1960. 10. T.H. Stix, Phys. Rev. L e t t e r s 15 (1965) 878. 11. J. Jan6af-lk et al., Generation and heating of p l a s m a by b e a m - p l a s m a interaction, P r o c . Third Conf. on P l a s m a p h y s i c s and controlled nuclear fusion Res. Novosibirisk, 1968, to be published.
OF
EINSTEIN
UNIVERSE
Z. H O t ~ K Department of Physics, Technical University, Praha, Czechoslovakia Received 12 August 1968
A closed relativistic s t e a d y - s t a t e model of the universe consisting of a very large number of oscillating "local u n i v e r s e s " is proposed. It c r e a t e s a nearly constant s c a l a r gravitational potential with average value - c 2, is in full a g r e e m e n t with Mach's principle and is not c o n t r a r y Lo astronomical observations.
T h e E i n s t e i n s t a t i c u n i v e r s e s e e m s not to b e a c c e p t a b l e a t p r e s e n t b e c a u s e of i t s i n s t a b i l i t y and disagreement with the observed red shift. M o r e o v e r , r e c e n t o b s e r v a t i o n s showed that the u n i v e r s e w a s v e r y m u c h d e n s e r in t h e p a s t t h a n it i s a t p r e s e n t . T h i s m e a n s t h a t t h e o b s e r v a b l e 332
p a r t of t h e u n i v e r s e d o e s not s a t i s f y t h e p e r f e c t cosmological principle. On t h e o t h e r h a n d , H o y l e a n d N a r l i k a r i n v e s t i g a t e d in 1965 [1] t h e c a s e in w h i c h t h e r e a r e d e p a r t u r e s f r o m h o m o g e n e i t y in t h e u n i v e r s e . T h e y found that t h e r e will be m a n y l o c a l i z e d finite o s -