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Parametric resonance in magnetosonic waves in plasma
Volume 25A, number 3
P H YS I C S L E T T E R S
r e c o m b i n a t i o n of a f r e e exciton in the n = 1 state, A1-LO i s the r e c o m b i n a t i o n of a free exciton with s i m u l t a n e o u s e m i s s i o n of a IX) phonon. The excitation light s o u r c e was provided by a u.v. l a s e r (ruby l a s e r followed by a KDP c r y s t a l ; a solution of SO4Cu a b s o r b s the r e d light a f t e r the KDP c r y s t a l ) . The v a r i a t i o n of the excitation i n t e n s i t y was obtained by m e a n s of f i l t e r s (the f i l t e r s c o n s i s t e d of v a r i o u s c o n c e n t r a t i o n s of CrO4K2 i n s i d e c a r e f u l l y s e l e c t e d plane p a r a l l e l g l a s s cells). The u.v. b e a m was focused with a c y l i n d r i c a l q u a r t z l e n s upon the s a m p l e . P a r t of the b e a m was sent upon a high speed photocell (CSF 9096 r i s e t i m e < 3 × 10-9s) in o r d e r to m e a s u r e the e x c i t a t i o n i n t e n s i t y . The e m i s s i o n was detected through a m o n o c h r o m a t o r (10 A / m m ) with a high speed PM (AVP56 r i s e t i m e ~ 2 × 10 . 9 s). Fig. 1 shows the r e s u l t s m e a s u r e d on the s a m e s a m p l e for A 1 and A1-LO for a d i r e c t i o n of p o l a r i z a t i o n of the e m i s s i o n p a r a l l e l to the c axis of the s a m p l e . (In this p o l a r i z a t i o n , line A 1 i s b e t t e r s e p a r a t e d f r o m bound exciton r e c o m b i nation lines). F o r a c e r t a i n excitation t h r e s h o l d , the v a r i a t i o n of the i n t e n s i t y of line A I b e c o m e s s u b l i n e a r when r e p r e s e n t e d on a d o u b l e - l o g a r i t h m i c s c a l e ; line A I - L O b e c o m e s s u p e r l i n e a r .
14 August 1967
Fig. 2 shows the dependence of A1-LO for E p e r p e n d i c u l a r to C. The p u l s e s of l u m i n e s c e n c e (left) and l a s e r (right) for the s u p e r l i n e a r a) and l i n e a r b) r e g i o n a r e also shown. The s h a r p i n c r e a s e in i n t e n s i t y of the phonon a s s i s t e d e m i s s i o n l i n e s is suggested to be due to the following m e c h a n i s m : photoexcited hot e l e c t r o n s r e l a x by e m i t t i n g LO phonons [2]; the phonons then s t i m u l a t e phonon a s s i s t e d e m i s s i o n . The dependence of bound excitons r e c o m b i n a tion l i n e s upon excitation i n t e n s i t y will be p r e s e n t e d in a following paper.
Refe~¢es 1. A. Mysyrowicz, J.B. Grun, F. Raga and S. Nikitine, Phys. Letters 24 (1967) 335. 2. D.W. Langer, Y.S.Park and R.N. Euwema, Phys. Rev. 152 (1966) 788.
* * * * *
PARAMETRIC
RESONANCE
IN MAGNETOSONIC
WAVES
IN PLASMA
D. A. FRANK-KAMENETSKII and Yu. N. SMIRNOV Kurchatov Atomic Energy Institute, Moscow
Received 3 July 1967
The analsyis of perturbed trajectories of particles in a modulated magnetic field shows that change of phase relations of particle motion and electrical field of the wave can !ead to the absorption of energy at the undertones of the cyclotron frequency.
In the theory of wave propagation in a p l a s m a the effects connected with the v a r i a b l e m a g n e t i c field of a wave a r e u s u a l l y taken into account only in higher a p p r o x i m a t i o n s of p e r t u r b a t i o n theory. The L o r e n t z force a s s o c i a t e d with this field i s u s u a l l y s m a l l . However, i m p o r t a n t effects may a r i s e f r o m quite d i f f e r e n t r e a s o n s . The m a g n e t i c field of the wave b r i n g s about a m o d u l a t i o n of the e x t e r n a l field. The modulated field affects the phase r e l a t i o n s between the p a r -
t i c l e motion and the e l e c t r i c field, and c o n s e q u e n t ly the energy a b s o r p t i o n f r o m the e l e c t r i c field. In such a way a p a r a m e t r i c r e s o n a n c e may occur. S i m i l a r p h e n o m e n a have been c o n s i d e r e d in [1,2] for a s i m p l i f i e d field g e o m e t r y with a l i n e a r l y i n c r e a s i n g induced e l e c t r i c field [see also 3, p. 301]. In the p r e s e n t p a p e r a m o r e r e a l i s t i c case of a plane wave i s c o n s i d e r e d , the d i r e c t i o n of the wave v e c t o r k being n o r m a l to the m a g n e t i c field. The u s u a l approach c o n s i s t s in finding out u n 287
Volume25A. number 3
PHYSICS LETTERS
p e r t u r b e d p a r t i c l e t r a j e c t o r i e s in an e x t e r n a l constant m a g n e t i c f i e l d / ' / , and in looking f o r c o r r e c t i o n s to t h e s e t r a j e c t o r i e s taking the wave f i e l d s as s m a l l p e r t u r b a t i o n s . On the c o n t r a r y , we shall look f o r p e r t u r b e d t r a j e c t o r i e s , i.e. f o r the t i m e dependence of the p a r t i c l e v e l o c i t y v in the f i e l d s of a plane wave c o n s i d e r i n g p r o p e r l y the t i m e dependence of f i e l d s , but a s s u m i n g that t h e i r v a r i a t i o n s in s p a c e should be s m a l l . The p r o b l e m is now r e d u c e d to the solution of the equation of motion f o r a p a r t i c l e with a m a s s M and a c h a r g e Z e : dv M~f=
ze (~ + ~ Iv xn]),
14 August 1967
Can be e x p r e s s e d in t e r m s of the B e s s e l function and contains r e s o n a n t t e r m s f o r each value of 5 -= Wc / ~o = n,
w h e r e n is an i n t e g e r . Only t h e s e r e s o n a n t t e r m s l e a d to e n e r g y a b s o r p t i o n . In the v i c i n i t y of the r e s o n a n c e condition (4) can be r e p l a c e d by 5 - n = (w c - n c o ) / w = E < 1.
(5)
Taking into account only t h e s e t e r m s we obtain the following e x p r e s s i o n f o r the p a r t i c l e e n e r g y in the random phase a p p r o x i m a t i o n :
(1) E ~ ~2
w h e r e the f i e l d s ~ and H depend a c t u a l l y on the v a r i a b l e k r - w t w h e r e r = R o + 6r is the r a d i u s v e c t o r of a p a r t i c l e . The p r o b l e m can be s o l v e d in p r i n c i p l e by s u c c e s s i v e a p p r o x i m a t i o n s taking 5 r f r o m the solution of eq. (1) in the p r e c e d i n g a p p r o x i m a t i o n . H e r e we adopt the s i m p l e s t a p p r o x i m a t i o n c o n s i d e r i n g long w a v e s or s m a l l d i s p l a c e m e n t s 6 r and r e g a r d i n g k. 5 r as a s m a l l p a r a m e t e r . Under this a s s u m p t i o n we c o m p l e t e l y n e g l e c t the s p a c e v a r i a t i o n of fields and u s e the f o l lowing e x p r e s s i o n s :
=E= cos(~t -~),
(4)
H= [Ho+H cos(~t-~)] . z . (2)
H e r e z is a unit v e c t o r in the d i r e c t i o n of a m a g n e t i c field, ¢o is the wave f r e q u e n c y , ~ --- k R is the i nit i al p h as e of the p a r t i c l e . The method can be a c t u a l l y used if the i n i ti a l p h a s e s m ay be taken as r a n d o m . T h i s i m p l i e s a r a n d o m t r a j e c t o r y shift at the d i s t a n c e 5 r due to c o l l i s i o n s o r o t h e r r e l a x a t i o n p r o c e s s e s . Then the condition of a p p l i c a b i l i t y of our m e t h o d ta k e s the f o r m (k, d 6 r / d t ) T ~ (k, v) ~ << 1. H e r e T i s the r e l a x a t i o n t i m e . Under t h e s e a s s u m p t i o n s the equation of motion (1) can be r e w r i t t e n in a c o m plex f o r m dw d-7-=~ a c o s ( w t - ~ ) -i¢0c [1 + ~ c o s @ t - ~ ) ] w ,
n
j _ n ( 6 h ) smeco~ewt
.
(6)
T h i s is a p e r i o d i c function; under the condition (5) the p e r i o d is long, and if the r e l a x a t i o n t i m e is s m a l l as c o m p a r e d to this p e r i o d , e n e r g y a b s o r p t i o n will take place. F o r the s t r i c t r e s o nance (6 =n) (6) tends to
t2 (ZeI l j. (Sh))2
E r e s ~ 2M \
h
(7)
In the l i m i t of h ~ 0 the r e s o n a n c e r e m a i n s only for 6 = 1, i.e. it t r a n s f o r m s into the usual c y c l o t r o n r e s o n a n c e , and eq. (7) t r a n s f o r m s into the well known e x p r e s s i o n f o r the e n e r g y a b s o r p tion at c y c l o t r o n r e s o n a n c e [3]
E - (Ze
)2 t2"
8M In the general case of non-zero modulation depth h the resonance frequencies are equal to the cyclotron frequency divided by an arbitrary integer (undertones). A detailed account of this work will be given in Zh. Eksp. i Teor. Fiz.
(3)
where
w =- v x + ivy;
¢oc =- Z e H o / M C
;
h =- H / H o ;
~ ( Z e / M ) (Ex + i E y ). A c c o r d i n g to r e f . 2, h is the modulation depth of the m a g n e t i c field. The e x a c t solution of eq.(3)
288
References 1. L.J.Rudakov and R.Z.Sagdeev, Plasma physics and the problem of controlled thermonuclear reactions, ed. M.A.Leontovich, (London, Pergamon Press, 1959). 2. Yu. N. Smirnovand D.A. Frank-Kamenetskii, Soviet physics, Doklady, Vol° 174, No. 6 (1967). 3. L.A.Artsimovich, Controlledthermonuclear reactions, (Oliverand Boyd, Edinburgh and London, 1964)
P H YS I C S L E T T E R S
r e c o m b i n a t i o n of a f r e e exciton in the n = 1 state, A1-LO i s the r e c o m b i n a t i o n of a free exciton with s i m u l t a n e o u s e m i s s i o n of a IX) phonon. The excitation light s o u r c e was provided by a u.v. l a s e r (ruby l a s e r followed by a KDP c r y s t a l ; a solution of SO4Cu a b s o r b s the r e d light a f t e r the KDP c r y s t a l ) . The v a r i a t i o n of the excitation i n t e n s i t y was obtained by m e a n s of f i l t e r s (the f i l t e r s c o n s i s t e d of v a r i o u s c o n c e n t r a t i o n s of CrO4K2 i n s i d e c a r e f u l l y s e l e c t e d plane p a r a l l e l g l a s s cells). The u.v. b e a m was focused with a c y l i n d r i c a l q u a r t z l e n s upon the s a m p l e . P a r t of the b e a m was sent upon a high speed photocell (CSF 9096 r i s e t i m e < 3 × 10-9s) in o r d e r to m e a s u r e the e x c i t a t i o n i n t e n s i t y . The e m i s s i o n was detected through a m o n o c h r o m a t o r (10 A / m m ) with a high speed PM (AVP56 r i s e t i m e ~ 2 × 10 . 9 s). Fig. 1 shows the r e s u l t s m e a s u r e d on the s a m e s a m p l e for A 1 and A1-LO for a d i r e c t i o n of p o l a r i z a t i o n of the e m i s s i o n p a r a l l e l to the c axis of the s a m p l e . (In this p o l a r i z a t i o n , line A 1 i s b e t t e r s e p a r a t e d f r o m bound exciton r e c o m b i nation lines). F o r a c e r t a i n excitation t h r e s h o l d , the v a r i a t i o n of the i n t e n s i t y of line A I b e c o m e s s u b l i n e a r when r e p r e s e n t e d on a d o u b l e - l o g a r i t h m i c s c a l e ; line A I - L O b e c o m e s s u p e r l i n e a r .
14 August 1967
Fig. 2 shows the dependence of A1-LO for E p e r p e n d i c u l a r to C. The p u l s e s of l u m i n e s c e n c e (left) and l a s e r (right) for the s u p e r l i n e a r a) and l i n e a r b) r e g i o n a r e also shown. The s h a r p i n c r e a s e in i n t e n s i t y of the phonon a s s i s t e d e m i s s i o n l i n e s is suggested to be due to the following m e c h a n i s m : photoexcited hot e l e c t r o n s r e l a x by e m i t t i n g LO phonons [2]; the phonons then s t i m u l a t e phonon a s s i s t e d e m i s s i o n . The dependence of bound excitons r e c o m b i n a tion l i n e s upon excitation i n t e n s i t y will be p r e s e n t e d in a following paper.
Refe~¢es 1. A. Mysyrowicz, J.B. Grun, F. Raga and S. Nikitine, Phys. Letters 24 (1967) 335. 2. D.W. Langer, Y.S.Park and R.N. Euwema, Phys. Rev. 152 (1966) 788.
* * * * *
PARAMETRIC
RESONANCE
IN MAGNETOSONIC
WAVES
IN PLASMA
D. A. FRANK-KAMENETSKII and Yu. N. SMIRNOV Kurchatov Atomic Energy Institute, Moscow
Received 3 July 1967
The analsyis of perturbed trajectories of particles in a modulated magnetic field shows that change of phase relations of particle motion and electrical field of the wave can !ead to the absorption of energy at the undertones of the cyclotron frequency.
In the theory of wave propagation in a p l a s m a the effects connected with the v a r i a b l e m a g n e t i c field of a wave a r e u s u a l l y taken into account only in higher a p p r o x i m a t i o n s of p e r t u r b a t i o n theory. The L o r e n t z force a s s o c i a t e d with this field i s u s u a l l y s m a l l . However, i m p o r t a n t effects may a r i s e f r o m quite d i f f e r e n t r e a s o n s . The m a g n e t i c field of the wave b r i n g s about a m o d u l a t i o n of the e x t e r n a l field. The modulated field affects the phase r e l a t i o n s between the p a r -
t i c l e motion and the e l e c t r i c field, and c o n s e q u e n t ly the energy a b s o r p t i o n f r o m the e l e c t r i c field. In such a way a p a r a m e t r i c r e s o n a n c e may occur. S i m i l a r p h e n o m e n a have been c o n s i d e r e d in [1,2] for a s i m p l i f i e d field g e o m e t r y with a l i n e a r l y i n c r e a s i n g induced e l e c t r i c field [see also 3, p. 301]. In the p r e s e n t p a p e r a m o r e r e a l i s t i c case of a plane wave i s c o n s i d e r e d , the d i r e c t i o n of the wave v e c t o r k being n o r m a l to the m a g n e t i c field. The u s u a l approach c o n s i s t s in finding out u n 287
Volume25A. number 3
PHYSICS LETTERS
p e r t u r b e d p a r t i c l e t r a j e c t o r i e s in an e x t e r n a l constant m a g n e t i c f i e l d / ' / , and in looking f o r c o r r e c t i o n s to t h e s e t r a j e c t o r i e s taking the wave f i e l d s as s m a l l p e r t u r b a t i o n s . On the c o n t r a r y , we shall look f o r p e r t u r b e d t r a j e c t o r i e s , i.e. f o r the t i m e dependence of the p a r t i c l e v e l o c i t y v in the f i e l d s of a plane wave c o n s i d e r i n g p r o p e r l y the t i m e dependence of f i e l d s , but a s s u m i n g that t h e i r v a r i a t i o n s in s p a c e should be s m a l l . The p r o b l e m is now r e d u c e d to the solution of the equation of motion f o r a p a r t i c l e with a m a s s M and a c h a r g e Z e : dv M~f=
ze (~ + ~ Iv xn]),
14 August 1967
Can be e x p r e s s e d in t e r m s of the B e s s e l function and contains r e s o n a n t t e r m s f o r each value of 5 -= Wc / ~o = n,
w h e r e n is an i n t e g e r . Only t h e s e r e s o n a n t t e r m s l e a d to e n e r g y a b s o r p t i o n . In the v i c i n i t y of the r e s o n a n c e condition (4) can be r e p l a c e d by 5 - n = (w c - n c o ) / w = E < 1.
(5)
Taking into account only t h e s e t e r m s we obtain the following e x p r e s s i o n f o r the p a r t i c l e e n e r g y in the random phase a p p r o x i m a t i o n :
(1) E ~ ~2
w h e r e the f i e l d s ~ and H depend a c t u a l l y on the v a r i a b l e k r - w t w h e r e r = R o + 6r is the r a d i u s v e c t o r of a p a r t i c l e . The p r o b l e m can be s o l v e d in p r i n c i p l e by s u c c e s s i v e a p p r o x i m a t i o n s taking 5 r f r o m the solution of eq. (1) in the p r e c e d i n g a p p r o x i m a t i o n . H e r e we adopt the s i m p l e s t a p p r o x i m a t i o n c o n s i d e r i n g long w a v e s or s m a l l d i s p l a c e m e n t s 6 r and r e g a r d i n g k. 5 r as a s m a l l p a r a m e t e r . Under this a s s u m p t i o n we c o m p l e t e l y n e g l e c t the s p a c e v a r i a t i o n of fields and u s e the f o l lowing e x p r e s s i o n s :
=E= cos(~t -~),
(4)
H= [Ho+H cos(~t-~)] . z . (2)
H e r e z is a unit v e c t o r in the d i r e c t i o n of a m a g n e t i c field, ¢o is the wave f r e q u e n c y , ~ --- k R is the i nit i al p h as e of the p a r t i c l e . The method can be a c t u a l l y used if the i n i ti a l p h a s e s m ay be taken as r a n d o m . T h i s i m p l i e s a r a n d o m t r a j e c t o r y shift at the d i s t a n c e 5 r due to c o l l i s i o n s o r o t h e r r e l a x a t i o n p r o c e s s e s . Then the condition of a p p l i c a b i l i t y of our m e t h o d ta k e s the f o r m (k, d 6 r / d t ) T ~ (k, v) ~ << 1. H e r e T i s the r e l a x a t i o n t i m e . Under t h e s e a s s u m p t i o n s the equation of motion (1) can be r e w r i t t e n in a c o m plex f o r m dw d-7-=~ a c o s ( w t - ~ ) -i¢0c [1 + ~ c o s @ t - ~ ) ] w ,
n
j _ n ( 6 h ) smeco~ewt
.
(6)
T h i s is a p e r i o d i c function; under the condition (5) the p e r i o d is long, and if the r e l a x a t i o n t i m e is s m a l l as c o m p a r e d to this p e r i o d , e n e r g y a b s o r p t i o n will take place. F o r the s t r i c t r e s o nance (6 =n) (6) tends to
t2 (ZeI l j. (Sh))2
E r e s ~ 2M \
h
(7)
In the l i m i t of h ~ 0 the r e s o n a n c e r e m a i n s only for 6 = 1, i.e. it t r a n s f o r m s into the usual c y c l o t r o n r e s o n a n c e , and eq. (7) t r a n s f o r m s into the well known e x p r e s s i o n f o r the e n e r g y a b s o r p tion at c y c l o t r o n r e s o n a n c e [3]
E - (Ze
)2 t2"
8M In the general case of non-zero modulation depth h the resonance frequencies are equal to the cyclotron frequency divided by an arbitrary integer (undertones). A detailed account of this work will be given in Zh. Eksp. i Teor. Fiz.
(3)
where
w =- v x + ivy;
¢oc =- Z e H o / M C
;
h =- H / H o ;
~ ( Z e / M ) (Ex + i E y ). A c c o r d i n g to r e f . 2, h is the modulation depth of the m a g n e t i c field. The e x a c t solution of eq.(3)
288
References 1. L.J.Rudakov and R.Z.Sagdeev, Plasma physics and the problem of controlled thermonuclear reactions, ed. M.A.Leontovich, (London, Pergamon Press, 1959). 2. Yu. N. Smirnovand D.A. Frank-Kamenetskii, Soviet physics, Doklady, Vol° 174, No. 6 (1967). 3. L.A.Artsimovich, Controlledthermonuclear reactions, (Oliverand Boyd, Edinburgh and London, 1964)