On description of interactions of collective oscillations in solids

Volume 33A, number 7

PHYSICS LETTERS

OF

14 December 1970

ON DESCRIPTION OF INTERACTIONS COLLECTIVE OSCILLATIONS IN S O L I D S G. I. SURAMLISHVILI

Tbilisi State University, Tbilisi, USSR Received 3 November 1970

We discuss a method to derive matrix elements determining the probability for wave processes in solids, with the participation of lattice oscillations.

A r e g u l a r method which allows to find m a t r i x e l e m e n t s f o r n o n - l i n e a r wave p r o c e s s e s with the p a r t i c i p a t i o n of l a t t i c e o s c i l l a t i o n s (a phonon field) is d e s c r i b e d in the p r e s e n t p a p e r . Such p r o c e s s e s a r e c a u s e d by the i n t e r a c t i o n of c a r r i e r p l a s m a with a phonon field. F o r b r e v i t y we r e s t r i c t o u r s e l v e s to the c a s e when t h e r e is no m a g n e t i c field; then the i n t e r a c t i o n is of a d e f o r m a t i o n c h a r a c t e r and the L a g r a n g i a n for the c o m b in a t io n of c a r r i e r p l a s m a and o s c i l l a t i o n f i e l d s may be w r i t t e n in the f o r m s :

L=~ffdxdvf~

~(v+Dr)2-emo(X+r)-e~o(x+r)+

A~(x+r)

+

Ct

+ ½fP [/~) 9'- C~ (17~)2]dx+~-S(Eo+E)2dx.

(1)

H e r e f a is the d i s t r i b u t i o n function of c a r r i e r s in the s t a t i o n a r y f i el d E o = - V ~ o , ~ is the deviation of the p o t e n t i a l f r o m q~o, ~ i s the l a t t i c e shift and r is the c a r r i e r shift in the fields (V~), E = -17~; A e x p r e s s e s the constant of i n t e r a c t i o n of the c a r r i e r s with the l a t t i c e ; ~= (V~), D = ~/~t+ (v V)+(e/m)(E o Vv). T he o t h e r s y m b o l s a r e conventional. When t h e r e is no phonon f i e l d (~= 0), eq. (1) g i v e s the Low L a g r a n g i a n [I]. Expanding in p o w e r s of r we r e p r e s e n t eq. (I) in the f o r m of a sum of L a g r a n g i a n s of d i f f e r e n t o r d e r s . Then the L a g r a n g i a n L 2 which is q u a d r a t i c in the p e r t u r b a t i o n ampltude d e s c r i b e s the n at u r al o s c i l l a t i o n s of the m e d i u m [I], while L a g r a n g i a n s of hi~gher o r d e r s d e s c r i b e i n t e r a c t i o n s b et w een n a t u r a l oscilla= t i ons [2-4]. To d e m o n s t r a t e the p r o c e d u r e of obtaining a m a t r i x e l e m e n t f r o m (i), let us c o n s i d e r a s p e c i f i c e x a m p l e : the decay of the / - p l a s m o n ( L a n g m u i r oscillation) to p r o c e d u r e an / - p l a s m o n and an a c o u s t i c phonon and the opposite p r o c e s s . Such p r o c e s s e s a r e d e s c r i b e d by a L a g r a n g i a n of the t h i r d o r d e r

L 3 = -½

~ff

d x d v f = r i o v i v j (e~o + A ( ) .

(2)

Let us r e p r e s e n t r a n d q~ in the f o r m r = r l + r 2 + r 3 , ~ = ~ l + q ~ 2 . The i n d i ces 1,2 denote p l a s m o n s and 3 is fo r phonons. The wave v e c t o r and the f r e q u e n c y of an l - p l a s m o n a r e denoted by k, ~ k r e s p e c t i v e l y ; the c o r r e s p o n d i n g q u a n t i t i e s f o r a phonon by q and wo. Let us expand r l 9 .~ ~I 9 in F o u r i e r s e r i e s , substitute t h e s e expansions in (2) and c o n s e r v e only the c r o s s t e r m s . Let us e x p r e s s F o u r i e r c o m p o n e n t s of the shifts r k , rq by F o u r i e r c o m p o n e n t s of the f i e l d s ~ k , ~q. We p r o c e e d f r o m the equation obtained by v a r i a t i o n of o v e r r : D 2 r = - ( l / m ) [ e 17¢+ A V(17~) ]. L e t us r e n o r m a l i z e q~k a c c o r d i n g to the equation:

"fL2dt

467

Volume 33A, number 7

~ 7 4~ k~l , 2 I~ ~- - |

PHYSICS

1

~ o)=~kl ,2 ~(1,212 ~Pkl'2 + ~pkl'2

LETTERS

= ~

kl, 2

/VkI

14 December 1970

'2~2k1'2

(3)

,

w h e r e El i s the l o n g i t u d i n a l d i e l e c t r i c p e r m e a b i l i t y of the m e d i u m . (3) s h o u l d b e c o n s i d e r e d a s the d e t e r m i n a t i o n of the d i s t r i b u t i o n f u n c t i o n N k of 1 - p l a s m o n s . T h e c o r r e s p o n d i n g e q u a t i o n f o r ~ q h a s the form

~pC2 q

q2~

w ~ . ,~ ~q = ~qn ~,~

Then introducing notation: K-1 4~akl,2 q~kl,2 = 1,2 {[ 8(¢o~l)/aw]w = £tkl,2

(4)

_1

akl'2'

~q=kpC2sq2 I

(5)

bq,

the L a g r a n g i a n (2) i s w r i t t e n in the f o r m :

L3 =

~

d)qk l k 2 bqaklak2

(6)

q+ k 1 + k 2 = 0

+ ak, b_q= b~, bq have the s e n s e of the c r e a t i o n a n d d e s t r u c t i o n o p e r a t o r s for T h e q u a n t i t i e s a_ k = ak, l - p l a s m o n s a n d p h o n o n s r e s p e c t i v e l y . If we u s e the c o n d i t i o n s ( k v ) << ~k ~ ~20, (qv) >> Wq, t h e n the f u n c t i o n $ q k l k 2 h a s the f o r m . q2 1/2 ~ i EOA ~ 3 / 2 ~2° (PCs)-I"/2 ( r n ( q ~ } q cos 9 , (7) qklk 2

~o

w h e r e ¢~ = c o + 2, ¢o i s the d i e l e c t r i c p e r m e a b i l i t y of the m e d i u m ( l a t t i c e ) , ~ o i s the p l a s m a 0 i s the a n g l e b e t w e e n k 1 a n d k 2 , ( ...... } m e a n s a v e r a g i n g o v e r the d i s t r i b u t i o n f u n c t i o n of T h e g e n e r a l i z a t i o n of the m e t h o d f o r the c a s e when t h e r e is an e x t e r n a l m a g n e t i c f i e l d is c u l t in p r i n c i p l e . C o n s i d e r a t i o n of a n u m b e r of s p e c i f i c p r o b l e m s with the u s e of m a t r i x e l e m e n t s o b t a i n e d a b o v e m e t h o d i s the c o n t e n t of a n o t h e r work. T h e a u t h o r e x p r e s s e s h i s g r a t i t u d e to D r . A. A. V e d e n o v for v a l u a b l e a d v i c e .

References [1] [2] [3] [4]

F . E . L o w , Proc. Roy. Soc. A248 (1958) 282. A. A. Vedenov, Problems of Plasma Theory, (Automizdat, issue 3, 1963). G.I. Suramlishvili, DokiadAkadNaukSSSR 153 {1963) 317. G. I. Suramlishvili, Zk. Eksp. i. Teor. Fiz. 52 {1967) 255.

468

frequency, carriers. not d i f f i by the