Dielectric constant of a weakly turbulent plasma

V o l u m e 29A, n u m b e r 5

PHYSICS

DIELECTRIC

CONSTANT

LETTERS

OF A WEAKLY

19 May 1969

TURBULENT

PLASMA

G. BERTHOMIEU

Institut d'Astrophysique , Paris, France Received 6 February 1969

The eigenfunctions of linearised Vlassov equation (Van Kampen modes) are used to calculate the dielectric constant of a weakly turbulent plasma and the correction to Langmuir frequency due to turbulence as a function of the energy of the excited waves.

The p r o p a g a t i o n of a weak amplitude wave of f r e q u e n c y to in a weakly t u r b u l e n t p l a s m a i s c o n s i d ered. In a t u r b u l a n t p l a s m a , the e l e c t r o s t a t i c e n e r g y in the excited modes is m o r e i m p o r t a n t than in the e q u i l i b r i u m state and it is closely r e l a t e d to the t w o - p a r t i c l e s c o r r e l a t i o n function. The p l a s m a will b e thus defined by its d i s t r i b u t i o n function and t w o - p a r t i c l e s c o r r e l a t i o n function at the i n i t i a l t i m e . T h e s e f u n c t i o n s r e l a x in a t i m e s c a l e of the o r d e r of ~ 1 (VK i s the Landau damping). We a s s u m e that the wave p e r i o d s a t i s f i e s the inequality 7K << w. The validity of the calculation is r e s t r i c t e d to wavelengths such that ~ 1 i s much s m a l l e r than the r e l a x a t i o n t i m e of the p l a s m a by b i n a r y c o l l i s i o n s , whose c o n t r i b u t i o n will b e thus neglected. The s t a r t i n g equations a r e the two f i r s t l i n e a r i s e d equations of BBGKY h i e r a r c h y , w h e r e the t h r e e p a r t i c l e s c o r r e l a t i o n s a r e neglected. An o r i g i n a l method of p r o j e c t i o n on the Van K a m p e n modes [ 1, 2], which a r e the eigenfunctions of the l i n e a r i s e d Vlassov equation, allow us to obtain in a s i m p l e m a n n e r the d i e l e c t r i c c o n s t a n t as a function of the i n i t i a l c o r r e l a t i o n state of the p l a s m a . Let AK(V, v ') be the F o u r i e r t r a n s f o r m of the t w o - p a r t i c l e s c o r r e l a t i o n function and AK(V , v'), AK(VV' ) i t s p r o j e c t i o n s on the Van K a m p e n modes, for example:

-oo

CKv Kv

The e x p r e s s i o n s of CKv and gKv can be found in ref. 2. The longitudinal d i e l e c t r i c c o n s t a n t c a l c u l a t e d to the f i r s t o r d e r in the r a t i o of Wt the t u r b u l e n t e n e r g y to Wc the k i n e t i c e n e r g y of the p l a s m a is: E(K,¢o) = ~ ° ( K , ¢ o ) I 1 - ~ 1 (a~O(K,v) "'1 ] rl=+X ¢o-kv rl \- ~ )v=v~K ~p(v~K)

K

w h e r e ~°(K, w) i s the u s u a l e x p r e s s i o n of the l i n e a r d i e l e c t r i c constant and v ~ a r e the r o o t s of E°(K, k v) +oo

~PK(V) = f f

dK'dv4Tre2 K'

m

~

1

k,2 av w - ( K - K ' ) v + k ' v

×

x {C(K,K', v, v) + , ( I f ' , v,v') T ( K ' , K ' - K , v ' , w ) f d v ' A K _ K , ( V ' , V ) + T ( K - K ' , - K ' , v , c o ) f d v ' A K , ( V , v')} + +f f f d K ' d v d v ' S ( K , IC ' v, v',v) co - IK-K'I 1 v - k ' v' P(K-K~'v'v') × × { T ( K _ K , , _ K , , v , , w ) f dv,,AK +K,(V,,V.) -

242

-k2"~v~Kv(K 'a ~

" ',v')}

Volume 29A, number 5

PHYSICS LETTERS

19May 1969

C(K,K', u,v) = 47re2 r K . mkk2 ~v AKt(V'V) + R(K', v,v') ~ . ~ - ~ A K _ K , ( V ' , v )1 4~e 2 I K '

T(K,K',v,~o) = - - - ~

~

0

1

K-K'

0 .,v)]

av ¢o-(K-K').v IK_K, 12"~v][ ]

S(K, K', v, v', v) = T(K, K', v, ¢o-k'v') + T(K, K-K', v, co' IK-K' {v) . The l i m i t i n g f o r m of e(K, w) f o r high f r e q u e n c i e s (¢o >> COp) can be c a l c u l a t e d f o r an i s o t r o p i c s p e c t r u m of t u r b u l e n c e and u s i n g a m o d e l for the d i s t r i b u t i o n function and c o r r e l a t i o n function. T h e c o r r e c tion AE (K, w) to the d i e l e c t r i c c o n s t a n t due to t u r b u l e n c e i s : Ac(K, 0~) ~ ~ ( W p ~ 4 Wt + i 2 5 (Wp.)3 Kv o k 2

\ w/

Wc

wt

\-'-w"/ Wp k 2 W c

w h e r e k2D i s the i n v e r s e s q u a r e Debye length, v o i s the t h e r m a l v e l o c i t y , COp is the p l a s m a f r e q u e n c y . At the l i m i t of cold p l a s m a , we obtain the c o r r e c t i o n Aw to the L a n g m u i r f r e q u e n c y due to t u r b u l e n c e , which i s of the s a m e o r d e r a s in r e f . 3:

Aw = Wp ~ f dK'W(g')(g mnwp

-Kt) 2 .

1. K.M. Case, Ann. Phys. 7 (1959) 349. 2. E. Asseo, G. Berthomieu, J. Heyvaerts and A. Mangeney, to be published. 3. L.M. Gorbunov and A. M. Tumerbulatov, Zh. Eksp. i Teor. Fiz. 26 (1968) 861. *****

ANISOTROPIC

EFFECTS

IN RADIATION

DAMAGE

OF

CRYSTALS

M. A. KUMAKHOV

Institute of Nuclear Physics, Moscow State University, Moscow, USSR Received 3 March 1969 A model of radiation cascade in crystals including ionization loss, atomic scattering anisotropy and anisotropy effects (channelling and focusing) is proposed.

The p u r p o s e of t h i s r e p o r t i s to e x a m i n e the r a d i a t i o n c a s c a d e t a k i n g into account o r d e r l i n e s s of c r y s t a l l a t t i c e , s c a t t e r i n g a n i s o t r o p y and ionization loss. T h e f o c u s i n g effect [ 1] of a t o m i c c o l l i s i o n s a s w e l l a s channelNug [2] should r e s u l t in a d e c r e a s e of p u r e l y a t o m i c d i s p l a c e m e n t s when Ef > E d (Ff i s the u l t i m a t e e n e r g y of focusing; E d i s the t h r e s h o l d e n e r g y of d i s p l a c e m e n t ) . In t h i s c a s e the i n c l u s i o n of f o c u s i n g i s equivalent to the inclusion~of the a n i s o t r o p y of the t h r e s h o l d e n e r g y of d i s p l a c e m e n t .

The p r o b a b i l i t y f o r focusing i s d e t e r m i n e d a s follows:

__

~~ ((a/D) ln E f / E +½)2

Wf ~ t (n. a/D) In g f / g

E ~E K ;

EK~Efexp

I°111 -~n

10~5

w h e r e n i s the p o s s i b l e n u m b e r of f o c u s i n g d i r e c t i o n s f o r the n u m b e r of d i s p l a c e m e n t s , V(E). In t h i s c a s e we obtain f o r the h a r d c o r e m o d e l

243