Echoes in collision-free plasmas

Volume25A. number 7

PHYSICS LETTERS

ECHOES

IN C O L L I S I O N - F R E E

9October 1967

PLASMAS

R. W. G O U L D

California Institute of Technology. Pasadena. California. USA Received 23 August 1967

It is shown that temporal echoes can occur in collision-free plasmas at various times for different wave number combinations and that spatial echoes can occur at various positions for different frequency combinations.

A p l a s m a wave echo in c o l l i s i o n - f r e e p l a s m a s has been p r e d i c t e d f r o m a p e r t u r b a t i o n solution of the V l a s o v equation [1]. H e r e we d e m o n s t r a t e the e x i s t e n c e of higher order t e m p o r a l as well as s p a t i a l e c h o e s , b as ed on a s i m p l e n o n p e r t u r b a tion method which g i v e s r e s u l t s to a l l o r d e r s . The s e l f - c o n s i s t e n t fie ld of the p l a s m a i s , however, neglected. We c o n s i d e r a p l a s m a of c h a r g e d p a r t i c l e s s u b j e c t e d to two exlernal e l e c t r i c f ie ld p u l s e s with d i f f e r e n t wave n u m b e r s at l= O, T, i.e., q E x / M = v 1 6(/) cos klX+ v 2 5(l-x) cos k2x. The position of a c h a r g e d p a r t i c l e whose i n i ti a l p o s i tion and v e l o c i t y is (Xo, u o) in such a f ie ld is r e a d i l y found to be x(t, Xo, Uo) = x o + + t(u o+ v 1 cos[klXo]) , f o r 0 -< 1 < r, and x(t, x o, u o) = x(7, Xo, Uo) + v 2 ( t - r) c o s [h2x(T,Xo,Uo) ] f or 7 ~< t < oo. The s p a ti a l F o u r i e r components of the c h a r g e density m a y be obtained by i n t e g r a t i n g o v e r the initial (Maxwellian) d i s t r i b u t i o n

Fo(Xo, Uo) , pk(t) =q f f Fo(xo, Uo) x x exp[-ikx(t, Xo, Uo)] dx o du o. Expanding the e x p o nential using well known B e s s e l function i d e n t i t i e s [2] and p e r f o r m i n g the i n t e g r a t i o n s , we find that Ok(1) v a n i s h e s unless

h = - m k 1 +nk 2 - k m n

(1)

w h e r e m and n a r e i n t e g e r s . Then we find that the c h a r g e density can be w r i t t e n Pk (t) = = noqAmn(l)gm n (t-Tmn), with

Tmn = ¢nk2/kmn)T, g m n q ) = ( 1 / m " ) ( k m n yt/2)m e -(kmn~l/2)2, Y m [ k m n V l ( t - Tmn)] A m n q ) = (_i)n-m [kmn v(t-Vrnn)/2] m / m ~ ×

(2) (3)

(4)

× Jn[kmn v2q-r)].

n o is the a v e r a g e density of c h a r g e d p a r t i c l e s and = ~ is t h e i r r . m . s , t h e r m a l speed. When v 1 and t' 2 a r e s m a l l c o m p a r e d with Y, Amn(t} is slowly v a r y i n g c o m p a r e d to gmn(l). F u r t h e r m o r e , only Amn(t) depends on the pulse a m p l i t u d e s t' 1 and v 2. In this l i m i t Amn(Vmn ) g i v e s the a m p l i lude of the pulse and gnm(l) g i v e s its shape. Thus we see that, p r o v i d e d 7ran > T, each sp at i al F o u r i e r component with combination wave n u m b er given by (1) exhibits a temporal echo at t i m e Trn n which is d e t e r m i n e d by the s e p a r a t i o n of the two applied p u l s e s and the r a t i o of t h e i r wave n u m b e r s . F o r weak p u l s e s (v 1 and v 2 s m a l l ) the amplitude r e d u c e s to A m n = (-i) n - m (Vl/~) m x x (v2/Y) n (mkl PT/2)n/n'., which exhibits the e x p e c t e d p o w e r law dependence on the pulse a m p l i tudes v 1 and v 2 and the pulse s e p a r a t i o n 7. A s i m i l a r a n a l y s i s has been made f o r sp at i al e c h o e s , which may be m o r e i m p o r t a n t e x p e r i m e n tally. When s p a t i a l l y l o c a l i s e d e l e c t r i c f i e l d s of d i f f er i n g f r e q u e n c i e s w1 and w2 a r e applied at z = 0 and z = l r e s p e c t i v e l y , in the p l a s m a , E x = = V1 5 (x) cos w 1 t + V2 5 (x-l) cos w 2t. The t e m p o r a l F o u r i e r components of the e l e c t r i c c u r r e n t d e n sity a r e found to vanish u n l e s s

w = -row 1 + nw 2 - Wren

(5)

and that each of t h e s e exhibits a l o c a l i s e d m a x i m u m at the position

x = (nw2/Wmn) 1 = l m n

(6)

p r o v i d e d that l m n > l. We a l s o find f o r the a m p l i tude Arnn(lmn) ~ [ ( m W l l / 2 ~ ) (e V 2 / n ~ 2 ) ] n x x [(nw21/2~ ) (e V 1/n~2)]rn when the d i m e n s i o n l e s s q u a n t i t i e s in the s q u a r e b r a c k e t s a r e s m a l l . E q s . (5) and (6) a r e the s p a t i a l an al o g s of e q s . (1) and (2), r e s p e c t i v e l y . Thus each t e m p o r a l F o u r i e r component of the e l e c t r i c c u r r e n t with f r e q u e n c i e s given by eq. (5) exhibits a spatial echo at a position which is d e t e r m i n e d by eq. (6). 559

Volume 25A. number 7

PHYSICS

In o b t a i n i n g t h e s e r e s u l t s the s e l f - c o n s i s t e n t f i e l d s of the p l a s m a h a v e b e e n n e g l e c t e d . T h i s a p p r o x i m a t i o n s h o u l d be v a l i d a s long as all w a v e n u m b e r s f o r which p k / p 0 is i m p o r t a n t a r e l a r g e (all f r e q u e n c i e s f o r which i k / i o is i m p o r t a n t a r e l a r g e ) . When t h e s e c o n d i t i o n s a r e not s a t i s f i e d we e x p e c t a s u b s t a n t i a l m o d i f i c a t i o n in the e c h o a m p l i t u d e s and s h a p e s . H o w e v e r , we s t i l l e x p e c t (2) [or (6)] to c o r r e c t l y g i v e the t i m e [or p o s i t i o n ] of the e c h o e s of the a p p r o p r i a t e w a v e n u m b e r (1) [or f r e q u e n c y (5)]. In a d d i t i o n , for s m a l l a m p l i t u d e s we s t i l l e x p e c t the p o w e r law d e p e n d e n c e on Vl, v 2 and T [or V1, V 2 and l] to hold. A m o r e c o m p l e t e t r e a t m e n t of t h i s p h e n o m e n o n is to a p p e a r [3].

LETTERS

9 October 1967

T h e a u t h o r is i n d e b t e d to T. M. O ' N e i l and J . H. M a l m b e r t f o r h e l p f u l d i s c u s s i o n s and to the O f f i c e of N a v a l R e s e a r c h f o r i t s s p o n s o r s h i p of this research.

Refcrenccs

1. R.W. Gould, T. M. O'Neil and J. H. Malmberg, Phys. Rev. Letters 19 {1967) 219. 2. W. Magnus and F. Oberhettinger, Special functions of mathematical physics, (Chelsea 1949, New York) p. 18. 20. 3. T. M. O'Neil and R.W. Gould, to be published.

SPINPOLARISATION DURCH ELEKTRONEN-RESONANZSTREUUNGAN NEON E. REICHERT und H. DEICHSEL Physikalisches Institut der Universitttl Mainz. Germany Eingegangen am 30 August 1967 Spinpolarization of 90°-electron-resonance scattering by neon-atoms is investigated in a double-scattering experiment. N a c h F r a n z e n und Gupta [1] s o U t e n u n t e r 0 = = 90 ° an Neon i n n e r h a l b d e s E n e r g i e b e r e i c h e s

d e r (P~, P±) - D o p p e l r e s o n a n z [2-4] (16.04 eV und 16.135 e V ) ~ g e s t r e u t e E l e k t r o n e n h o c h g r a d i g p o l a r i

5TREUSTUFE "IT

STREUSTUFE I Monochromator

Detektor

/

{

~ -~ ,~

WEST

Nachbeschieunigung

1111111 a U

Detektor I~ORD

300eV

1, % Eleklronensumpf ~ / >

I

Detektor OST

K~fig OST

Fig. 1. Prinzip der Versuchsanordnung; nicht massstabsgerecht. Der Monochromator und alle Elemente in der unmittelbaren Umgebung des Streuraumes in Streustufe 1 bestehen aus graphitiertem und k e r a m i k - i s o l i e r t e n Kupfer und werden bei 450°C ausgeheizt. Das Ausheizen der Gesamtapparatur is nicht erforderlieh. Kiifig WEST oder OST und Detektor NORD werden w~hrend des Messung als Monitor benutzt. 560