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Observation of surface modes excited by an electron beam in a plasma
Volume 29A, number 6
P H Y S I C S L E TT E R S
2 June 1969
2. H. Dammann, Synthetitsche Phasenhologramme, Annual Meeting of the Deutsche Gesellschaft ftlr angewandte Optik, Ltibeck, May 1969, to be published.
References 1. J. c. Urbach and R. W. Meier, Holographic recording materials, SPIE Holography Seminar Proc. Vol. 15, San Francisco, Calif., 1968. * * * * *
OBSERVATION OF SURFACE BY AN ELECTRON BEAM
MODES EXCITED IN A PLASMA *
H. BOHMER and M. R A E T H E R Coordinated Science Laboratory and Department of Physics. University of Illinois, Urbana, Illinois 61801, USA Received 17 April 1969
Unstable oscillations at frequencies below the plasma frequency are observed at the plasma boundary during the interaction of an electron beam with a plasma. They are interpreted as the eigenmodes of the inhomogeneous plasma boundary, driven into instability by the beam.
To study the s p at i a l d e v e l o p m e n t of w a v e s in a b e a m - p l a s m a system, a pulsed electron beam, 0 . 1 t o 1 A , 1 8 k V , 3 p s e c d u r a t i o n and 1 c m d i a m e t e r i s i n j e c t e d into a d e c a y i n g neon a f t e r glow p l a s m a , contained in a P y r e x tube 4 cm in d i a m e t e r and 40 c m long. No e x t e r n a l m a g n e t i c f i e l d is applied to the s y s t e m . The e x p e r i m e n t is p e r f o r m e d with a r e p e t i t i o n f r e q u e n c y of 15 Hz. Th e e l e c t r o n b e a m can be e x p o s e d to d i f f e r e n t p l a s m a d e n s i t i e s by changing the t i m e d e lay b e t w e e n the p l a s m a and the b e a m pulse. A m i c r o w a v e r e c e i v e r (26.5 to 40 GHz, 11 dB n o i s e f i g u re) r e c e i v e d the m i c r o w a v e p o w e r r a diated by the instability. U s i n g a m i c r o w a v e l e n s , the s p at i al r e s o l u t i o n i s a p p r o x i m a t e l y 1 cm. T h e p o l a r i z a t i o n of the r e c e i v i n g horn is such that the E - v e c t o r is p a r a l l e l to the e l e c t r o n b e a m . F o r each p o s i t i o n of the r e c e i v e r along the a x i s of the tube the r a d i a t e d p o w e r is r e c o r d e d a s a function of p l a s m a density. T h e i n s e r t of fig. 1 shows s o m e s a m p l e s of such r e c o r d i n g s f o r s e v e r a l p o s i t i o n s of the r e c e i v e r . Two f r e q u e n c y g r o u p s a r e o b s e r v e d . At long d i s t a n c e s f r o m the e n t r a n c e a p e r t u r e , the s i g n a l o c c u r s when the r e c e i v e r f r e q u e n c y is equal to the p l a s m a f r e q u e n c y (fig. 1, signal #1). T h i s is r a d i a t i o n f r o m the b e a m - p l a s m a i n s t a b i l i t y which h as the m a x i m u m growth r a t e at a * This work was supported by the Joint Services Electronics Program (U.S. Army, U.S. Navy, and U.S. Air Force) under contract DAAB07-67-C-O199. 302
f r e q u e n c y c l o s e to the p l a s m a f r e q u e n c y [ 1]. It e x h i b i t s the e x p e c t e d exponential i n c r e a s e with d i s t a n c e [2, 3]. At s m a l l e r d i s t a n c e s a signal that c o n s i s t s of s e v e r a l p e a k s and b e c o m e s s t r o n g e r with d e c r e a s i n g d i s t a n c e is r e c e i v e d at d el ay t i m e s w h e r e the r e c e i v e r f r e q u e n c y is much s m a l l e r than the p l a s m a f r e q u e n c y (fig. 1, signal #2a,b, c). F r o m the m ag n i t u d e of the r a d i a t e d p o w e r we conclude that, f o r e x a m p l e , at z = 1 c m , the e n e r g y in the e l e c t r i c f i el d f l u c t u a t i o n s was i n c r e a s e d by a f a c t o r of 105 to 106 above the t h e r m a l e q u i l i b r i u m value. Th e s a m e r e s o n a n c e s , although with a much s m a l l e r a m p l i t u d e , a r e o b s e r v e d at the opposite end of the d i s c h a r g e tube, i . e . , at the point w h e r e the b e a m l e a v e s the plasma. In s t a r t i n g the d i s c u s s i o n about the o r i g i n of t h e s e m o d e s , we h a v e to note that 1) no m o d u l a tion could be d e t e c t e d on the b e a m b e f o r e it e n t e r e d the p l a s m a , and 2) the change in the e n e r gy d i s t r i b u t i o n function of the e l e c t r o n b e a m due to the excitation of t h e s e w a v e s w as l e s s than
0.3%. Since we p e r f o r m the e x p e r i m e n t in a d e c a y ing p l a s m a , a r a d i a l as well a s ax i al d e n s i t y g r a d i e n t will b e p r e s e n t , the l a t t e r only c l o s e to the ends of the d i s c h a r g e tube. The b e a m has to t r a v e r s e t h e s e a x i a l d en si t y g r a d i e n t s in e n t e r ing and l e a v i n g the p l a s m a . We would like to s u g g e s t that the o s c i l l a t i o n s o b s e r v e d at s m a l l z a r e the e i g e n m o d e s of the i n h o m o g e n e o u s p l a s m a
PHYSICS LETTERS
Volume 29A, number 6 I
4l
I
2 =(~IT(m+~)kD/L)~3 c~2/~JpL
Distc,ire from Entrance Aperture
Signal No.
2(0) (b) (c)
Receiver Attenuation
12.0 cm
L
j,
I
103
16dB
' ' J 1
70
0
4.5
lO
1.0
25)~
e~
o
16o
2( 4 0 3 5 3027
I
10 2
) Delaylime {p.se
- Plasma freq. 7 (GHz) o /
-I I
26.SGHz = Rec. freq.
o: lc0-
%%% 1000
2
4
6
8
Distance from Entrance Aperture
10
12
(cm)
Fig. 1. Received power versus delay time and plasma frequency (measured independently with a 37 GHz m i crowave inteferometer) for several positions of the receiver and received power versus distance for three resonances: 1) (~ =OJrec = ~ u ; 2a) and ~h) (e = ~ r e c < < ~Jp. Receiver frequency: 2B.5 GHz; electron beam: 170 mA, 18 kV ; pressure: 54 /~ neon. b o u n d a r y d r i v e n into i n s t a b i l i t y by the e l e c t r o n beam. It i s well known that in an inhomogeneous p l a s m a , s u r f a c e waves can exist [4]. The c a s e of a s e m i - i n f i n i t e p l a s m a has been d i s c u s s e d by Hoh [5], who t r e a t s this p r o b l e m a s s u m i n g a d e n s i t y p r o f i l e given by n = n L z / L . He finds for the mode f r e q u e n c i e s for a n o n - m a g n e t i z e d p l a s ma
2 June 1969
m =0,1,2 ....
w h e r e WpL and n L a r e the p l a s m a f r e q u e n c y and d e n s i t y at the point z = L . XD i s the Debye length. Since X D / L << 1 for m o s t e x p e r i m e n t s , the f r e q u e n c y of t h e s e modes will b e s m a l l e r than ~pL- In the p r e s e n c e of an e l e c t r o n b e a m one may expect, i n analogy to the infinite b e a m p l a s m a s y s t e m , that i n s t a b i l i t i e s o c c u r at f r e q u e n c i e s close to the eigenmodes. Since t h e s e waves exist only i n the region of finite d e n s i t y g r a d i e n t they will not exhibit a convective growth. The o b s e r v e d f r e q u e n c i e s do not c o incide exactly with the r e s o n a n c e s c a l c u l a t e d by Hoh for any value of XD/L, o r with those c a l c u lated f r o m the m o r e r e a l i s t i c p r o f i l e n ( z ) = = n o (1 - e x p ( - z / L ) ) . This d i s c r e p a n c y i s not too s u r p r i s i n g b e c a u s e the intuitive p i c t u r e of a standing wave between the b o u n d a r y and the plane w h e r e w = ~V will lose its validity i n the p r e s e n c e of a beavh s i n c e the b e a m - p l a s m a s y s t e m can support waves of f r e q u e n c i e s l e s s than .~p that a r e located on the u n s t a b l e b r a n c h of the dispersion relation. The development of such a n i n s t a b i l i t y c l o s e ly r e s e m b l e s a case t r e a t e d t h e o r e t i c a l l y by F a i n b e r g and Shapiro [3] who i n v e s t i g a t e d the t e m p o r a l and spatial development of an i n s t a b i l ity u n d e r the hypothetical condition of z e r o group velocity of the u n s t a b l e waves. They found that the s y s t e m i s u n s t a b l e and that the o s c i l l a t i o n s a c c u m u l a t e i n a n a r r o w l a y e r close to the p l a s m a b o u n d a r y in a s i m i l a r f a s h i o n as o b s e r v e d in our m e a s u r e m e n t s .
References 1. v . s. Imshennik and Yu. I. M o r o z o v , Sov. P h y s . Techn. Phys. 6 (1961) 464. 2. B . B . Kadomtsev, Plasma turbulence (Academic Press, London, 1965). 3. Ya. B. Fainberg and V. D. Shapiro, Soviet Phys. J E T P 20 (1965) 937. 4. A. Dattner, Phys. Rev. L e t t e r s 10 (1963) 205;
T. V. Parker, J, C. Nickel and R. Gould, Phys. Fluids 7 (1964) 1489; P.Weissglas, J. Nucl. Energy, Pt. C6 (1964) 251; F. W. Crawford, J. Appl. Phys. 35 (1964) 1365. 5. F. C. Hoh, Phys. P~v. 133 (1964) A1016.
303
P H Y S I C S L E TT E R S
2 June 1969
2. H. Dammann, Synthetitsche Phasenhologramme, Annual Meeting of the Deutsche Gesellschaft ftlr angewandte Optik, Ltibeck, May 1969, to be published.
References 1. J. c. Urbach and R. W. Meier, Holographic recording materials, SPIE Holography Seminar Proc. Vol. 15, San Francisco, Calif., 1968. * * * * *
OBSERVATION OF SURFACE BY AN ELECTRON BEAM
MODES EXCITED IN A PLASMA *
H. BOHMER and M. R A E T H E R Coordinated Science Laboratory and Department of Physics. University of Illinois, Urbana, Illinois 61801, USA Received 17 April 1969
Unstable oscillations at frequencies below the plasma frequency are observed at the plasma boundary during the interaction of an electron beam with a plasma. They are interpreted as the eigenmodes of the inhomogeneous plasma boundary, driven into instability by the beam.
To study the s p at i a l d e v e l o p m e n t of w a v e s in a b e a m - p l a s m a system, a pulsed electron beam, 0 . 1 t o 1 A , 1 8 k V , 3 p s e c d u r a t i o n and 1 c m d i a m e t e r i s i n j e c t e d into a d e c a y i n g neon a f t e r glow p l a s m a , contained in a P y r e x tube 4 cm in d i a m e t e r and 40 c m long. No e x t e r n a l m a g n e t i c f i e l d is applied to the s y s t e m . The e x p e r i m e n t is p e r f o r m e d with a r e p e t i t i o n f r e q u e n c y of 15 Hz. Th e e l e c t r o n b e a m can be e x p o s e d to d i f f e r e n t p l a s m a d e n s i t i e s by changing the t i m e d e lay b e t w e e n the p l a s m a and the b e a m pulse. A m i c r o w a v e r e c e i v e r (26.5 to 40 GHz, 11 dB n o i s e f i g u re) r e c e i v e d the m i c r o w a v e p o w e r r a diated by the instability. U s i n g a m i c r o w a v e l e n s , the s p at i al r e s o l u t i o n i s a p p r o x i m a t e l y 1 cm. T h e p o l a r i z a t i o n of the r e c e i v i n g horn is such that the E - v e c t o r is p a r a l l e l to the e l e c t r o n b e a m . F o r each p o s i t i o n of the r e c e i v e r along the a x i s of the tube the r a d i a t e d p o w e r is r e c o r d e d a s a function of p l a s m a density. T h e i n s e r t of fig. 1 shows s o m e s a m p l e s of such r e c o r d i n g s f o r s e v e r a l p o s i t i o n s of the r e c e i v e r . Two f r e q u e n c y g r o u p s a r e o b s e r v e d . At long d i s t a n c e s f r o m the e n t r a n c e a p e r t u r e , the s i g n a l o c c u r s when the r e c e i v e r f r e q u e n c y is equal to the p l a s m a f r e q u e n c y (fig. 1, signal #1). T h i s is r a d i a t i o n f r o m the b e a m - p l a s m a i n s t a b i l i t y which h as the m a x i m u m growth r a t e at a * This work was supported by the Joint Services Electronics Program (U.S. Army, U.S. Navy, and U.S. Air Force) under contract DAAB07-67-C-O199. 302
f r e q u e n c y c l o s e to the p l a s m a f r e q u e n c y [ 1]. It e x h i b i t s the e x p e c t e d exponential i n c r e a s e with d i s t a n c e [2, 3]. At s m a l l e r d i s t a n c e s a signal that c o n s i s t s of s e v e r a l p e a k s and b e c o m e s s t r o n g e r with d e c r e a s i n g d i s t a n c e is r e c e i v e d at d el ay t i m e s w h e r e the r e c e i v e r f r e q u e n c y is much s m a l l e r than the p l a s m a f r e q u e n c y (fig. 1, signal #2a,b, c). F r o m the m ag n i t u d e of the r a d i a t e d p o w e r we conclude that, f o r e x a m p l e , at z = 1 c m , the e n e r g y in the e l e c t r i c f i el d f l u c t u a t i o n s was i n c r e a s e d by a f a c t o r of 105 to 106 above the t h e r m a l e q u i l i b r i u m value. Th e s a m e r e s o n a n c e s , although with a much s m a l l e r a m p l i t u d e , a r e o b s e r v e d at the opposite end of the d i s c h a r g e tube, i . e . , at the point w h e r e the b e a m l e a v e s the plasma. In s t a r t i n g the d i s c u s s i o n about the o r i g i n of t h e s e m o d e s , we h a v e to note that 1) no m o d u l a tion could be d e t e c t e d on the b e a m b e f o r e it e n t e r e d the p l a s m a , and 2) the change in the e n e r gy d i s t r i b u t i o n function of the e l e c t r o n b e a m due to the excitation of t h e s e w a v e s w as l e s s than
0.3%. Since we p e r f o r m the e x p e r i m e n t in a d e c a y ing p l a s m a , a r a d i a l as well a s ax i al d e n s i t y g r a d i e n t will b e p r e s e n t , the l a t t e r only c l o s e to the ends of the d i s c h a r g e tube. The b e a m has to t r a v e r s e t h e s e a x i a l d en si t y g r a d i e n t s in e n t e r ing and l e a v i n g the p l a s m a . We would like to s u g g e s t that the o s c i l l a t i o n s o b s e r v e d at s m a l l z a r e the e i g e n m o d e s of the i n h o m o g e n e o u s p l a s m a
PHYSICS LETTERS
Volume 29A, number 6 I
4l
I
2 =(~IT(m+~)kD/L)~3 c~2/~JpL
Distc,ire from Entrance Aperture
Signal No.
2(0) (b) (c)
Receiver Attenuation
12.0 cm
L
j,
I
103
16dB
' ' J 1
70
0
4.5
lO
1.0
25)~
e~
o
16o
2( 4 0 3 5 3027
I
10 2
) Delaylime {p.se
- Plasma freq. 7 (GHz) o /
-I I
26.SGHz = Rec. freq.
o: lc0-
%%% 1000
2
4
6
8
Distance from Entrance Aperture
10
12
(cm)
Fig. 1. Received power versus delay time and plasma frequency (measured independently with a 37 GHz m i crowave inteferometer) for several positions of the receiver and received power versus distance for three resonances: 1) (~ =OJrec = ~ u ; 2a) and ~h) (e = ~ r e c < < ~Jp. Receiver frequency: 2B.5 GHz; electron beam: 170 mA, 18 kV ; pressure: 54 /~ neon. b o u n d a r y d r i v e n into i n s t a b i l i t y by the e l e c t r o n beam. It i s well known that in an inhomogeneous p l a s m a , s u r f a c e waves can exist [4]. The c a s e of a s e m i - i n f i n i t e p l a s m a has been d i s c u s s e d by Hoh [5], who t r e a t s this p r o b l e m a s s u m i n g a d e n s i t y p r o f i l e given by n = n L z / L . He finds for the mode f r e q u e n c i e s for a n o n - m a g n e t i z e d p l a s ma
2 June 1969
m =0,1,2 ....
w h e r e WpL and n L a r e the p l a s m a f r e q u e n c y and d e n s i t y at the point z = L . XD i s the Debye length. Since X D / L << 1 for m o s t e x p e r i m e n t s , the f r e q u e n c y of t h e s e modes will b e s m a l l e r than ~pL- In the p r e s e n c e of an e l e c t r o n b e a m one may expect, i n analogy to the infinite b e a m p l a s m a s y s t e m , that i n s t a b i l i t i e s o c c u r at f r e q u e n c i e s close to the eigenmodes. Since t h e s e waves exist only i n the region of finite d e n s i t y g r a d i e n t they will not exhibit a convective growth. The o b s e r v e d f r e q u e n c i e s do not c o incide exactly with the r e s o n a n c e s c a l c u l a t e d by Hoh for any value of XD/L, o r with those c a l c u lated f r o m the m o r e r e a l i s t i c p r o f i l e n ( z ) = = n o (1 - e x p ( - z / L ) ) . This d i s c r e p a n c y i s not too s u r p r i s i n g b e c a u s e the intuitive p i c t u r e of a standing wave between the b o u n d a r y and the plane w h e r e w = ~V will lose its validity i n the p r e s e n c e of a beavh s i n c e the b e a m - p l a s m a s y s t e m can support waves of f r e q u e n c i e s l e s s than .~p that a r e located on the u n s t a b l e b r a n c h of the dispersion relation. The development of such a n i n s t a b i l i t y c l o s e ly r e s e m b l e s a case t r e a t e d t h e o r e t i c a l l y by F a i n b e r g and Shapiro [3] who i n v e s t i g a t e d the t e m p o r a l and spatial development of an i n s t a b i l ity u n d e r the hypothetical condition of z e r o group velocity of the u n s t a b l e waves. They found that the s y s t e m i s u n s t a b l e and that the o s c i l l a t i o n s a c c u m u l a t e i n a n a r r o w l a y e r close to the p l a s m a b o u n d a r y in a s i m i l a r f a s h i o n as o b s e r v e d in our m e a s u r e m e n t s .
References 1. v . s. Imshennik and Yu. I. M o r o z o v , Sov. P h y s . Techn. Phys. 6 (1961) 464. 2. B . B . Kadomtsev, Plasma turbulence (Academic Press, London, 1965). 3. Ya. B. Fainberg and V. D. Shapiro, Soviet Phys. J E T P 20 (1965) 937. 4. A. Dattner, Phys. Rev. L e t t e r s 10 (1963) 205;
T. V. Parker, J, C. Nickel and R. Gould, Phys. Fluids 7 (1964) 1489; P.Weissglas, J. Nucl. Energy, Pt. C6 (1964) 251; F. W. Crawford, J. Appl. Phys. 35 (1964) 1365. 5. F. C. Hoh, Phys. P~v. 133 (1964) A1016.
303