A THEORY OF THE INTERACTION OF DOUBLE-BEAM WITH PLASMA

HAL Id: jpa-00219374

https://hal.archives-ouvertes.fr/jpa-00219374

Submitted on 1 Jan 1979

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A THEORY OF THE INTERACTION OF DOUBLE-BEAM WITH PLASMA

A. Rogashkova, T. Shatalova

To cite this version:

A. Rogashkova, T. Shatalova. A THEORY OF THE INTERACTION OF DOUBLE- BEAM WITH PLASMA. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-781-C7-782.

�10.1051/jphyscol:19797377�. �jpa-00219374�

JOURiv'AL DE PHYSIQUE CoZZoque C7, suppZ6ment au n07, Tome 40, JuiZZet 1979, nage C7- 781

A THEORY OF THE INTERACTION OF DOUBLE-BEAM W I T H PLASMA

A.I. Rogashkova, T.I. Shatalova.

I n s t i t u t e o f Radioengineering and EZectronics, Academy o f Sciences, Moscow, U.S.S.R.

1. I n [I] i s e x p e r i m e n t a l l y shown t h e i n s t a b i l i t y development a r e c o n s i d e r a b l y a f f e c t e d by a n i n j e c t i o n of a second beam

A t h e o r e t i c a l i n v e s t i g a t i o n of t h e do- uble-beam i n t e r a c t i o n w i t h plasma i s pre-

s e n t e d in t h i s paper i n t h e l i n e a r and n o n l i n e a r regimes,

Two e l e c t r o n beams w i t h t h e c o n s t a n t v e l o c i t i e s Wb, and I?& a r e assumed t o mo- ve a l o n g a n a x i s of a c y l i n d r i c a l n e t a l - l i c waveguide f i l l e d by a cold homogene- ous plasma. The c r o s s d i s p l a c e m e n t s i s supposed t o a b s e n t ,

1I.Suppose t h a t t h e r a d i u s e s of t h e beams a r e e q u a l t o t h e r a d i u s of t h e wave guide, The d i s p e r s i o n e q u a t i o n d i s c r i b i n g t h e i n t e r a c t i o n of two e l e c t r o n beams in a plasma waveguide in t h e q u a s i s t a t i c ap- preachment i s

(1) where U -the r a d i u s of t h e vraveguide

,

%(P,)=O,&z&z- t h e a x i a l and c r o s s plasma per-

(J

-

a frequency,wp,c+,g~z- t h e plasma e l e - c t r o n f r e q u e n c i e s of plasma and beams r e - s p e c t i v e l y ,

-

an a x i a l wave number,

(A?,,

-

a gyrofrequency.

The d i s p e r s i o n e q u a t i o n (I; was c a l - c u l a t e d by t h e computer r e l a t i n g on (ra )

*

f o r d i f f e r e n t va- in dependence on

s=-

_{* P}

l u e s of t h e parameters:

bd,,2=

^C*l,gdz

uor, 2

~ 2 = ~ , 5 ; h z 4 2 ;

o,,=o.

UP GP

I n Fig. I a r e r e p r e s e n t e d dependences Reb+

and

Ymp

on t h e normalized frequency

f o r r,,&=l

,

b)pgI?=2,Z.

A s i s e v i d e n t from Fig.1 t h e beams i n t e - r a c t w i t h plasma and d o n ' t i n t e r a c t w i t h each o t h e r in r e g i o n c. 0,8.

I f 5 > 0,8 t h e branch

Red"

of an each beam i s s p l l i t t e d i n t o 1 . ~ 0 branches t h a t corresponds t o the f a s t and slow space charge waves ( f ,s.c.w., s.s.c.w).

Im

r e - , gion 0 , 9 < F L I ,6 t h e branches d e s c r i b i n g f.s.cw and s.s.c.w of d i f f e r e n t beams a r e connected in one branch, i.e. the double- beam i n t e r a c t i o n t a k e s p l a c e .

In Pig. 2 t h e c u r v e s of type curve 2 on Pig.? a r e p l o t t e d f o r t h e folbowing v a l u e s parameters:

rp,

^{= I ; I- =I , 2 5 ;}

11

- rP2

~ 2 9 7 5 ; 111

- 'GZ

^=6,75.

The d o t t e d c u r v e s d e s c r i b e d t h e a m p l i f i - c a t i o n r a t e s in the s p e c i a l case up =O f o r t h e same parameters

rp,

_&a

From t h e

9rnp-T

diagram i t f o l l o w s that in our system t h e a m p l i f i c a t i o n i s d e s c r i b e d by t h e two curves. The curve 7 h a s t h e same dependence c h a r a c t e r on a s in c a s e of t h e i n t e r a c t i o n of one beam w i t h plasma. The curve 2 due t o presen-

ce of a second beam shows t h e a m p l i f i c a - t i o n i s p o s s i b l e on t h e f r e q u e n c i e s ex- ceeding a plasma frequency. Thus t h e abo- ve r e s u l t s show that i n j e c t i o n of a se- cond beam i n t o plasma g r e a t l y changes t h e d i s p e r s i o n c h a r a c t e r i s t i c s of t h e plasrna- beam system.

111. Let u s c o n s i d e r OH =O when t h e slow waves cannot propagate a l o n g t h e wa- ve-guide i n absence of beams. It i s sup- posed t h a t u p B d , p c < up and t h e n o n l i - n e a r e f f e c t s in t h e system a r e only due

t o beams. I n plane Z =O t h e beams a r e modulated by a s i g n a l w i t h t h e e q u i d i -

s t a n t f r e q u e n c i e s w = m

.

T h e s y s t e m of e q u a t i o n s d e s c r i b i n g a space e v o l u t i o n of t h e waves e x c i t e d by one beam in t h e plasma waveguide i s g i v e n in

121 .

^Gene-

r a l i s i n g i t i n case of two beams we ob-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797377

t a p :

5 gn,ln, ⁺

a

9f121nl ]e~n@J;21

&$ ^i- (!+ STR~

^n-

²¹

^{nlr +dl [j+dl n}

-

-

, - .

%= nt,

⁰

to- an e n t r a n c e moment of p a r t i c l e s i n - t o s p a c e of i n t e r a c t i o n ;

@=br,% _-

t h e normalized d i s t a n c e ; &r,a

-

t h e dimen- s i o n l e s s amplitude of c u r r e n t on t h e f r e - quency "2;

In= in/k,

d = &/pi,

I ' d 8

h z ^{i2im4' ;}

i n -

t h e amplitude of c u r r e n t on t h e frequency nJZ ;

1,-

t h e c o n s t a n t component of c u r r e n t ; f,

-

^{t h e}

electron-beam-plasma frequency r e d u c t i o n f a c t o r [ 2 3 ;

)Qn -

t h e d i m e n s i o n l e s s am- p l i t u d e a x i a l component of e l e c t r i c a l f i - e l d on t h e frequency

nn:

j; &

(ydn""S n mOuo

The system ( 2 ) was ca1;ulated by t h e computer f o r [ l o o p a r t i c l e s of every beam.

It was solved under t h e f o l l o w i n g bounda- rs c o n d i t i o n s :

- ^-

To demonstrate"the i n f l u e n c e of t h e s e cond beam on t h e space e v o l u t i o n and spe- ctrum of the e x c i t e d waves t h e dependence of amplitude

I,,

of t h e f i r s t beam f o r t h e d i f f e r e n t v a l u e s of t h e parameters

1 :

'

and d i s shown i n Fig.3. The c a l - c u l a t i o n s were hold f o r t h e c a s e (GI:& /

/@$tf

)=0,0l and

P,

Irl =O;

en,

^=-0,I;

-0,2;-0,3;0,2;0,1. From t h e a n a l y s i s of t h e s e c u r v e s i t f o l l o w s t h a t bunching t h e - e l e c t r o n s of t h e f i r s t beam under t h e i n - f l u e n c e of t h e second one from some d i s -

t a n c e t a k e s p l a c e and t h e amplitude of harmonics of t h e c u r r e n t

In(

i n c r e a s e s i n t h e space. The bunching i~iill be more dense i f t h e second beam moves q u i c k e r than t h e f i r s t one (See Fig.3a7b) The amplitude i s most, because and **

If,/

i s t h e l a r g e s t /2]

.

^{I n} F i g . 3 ~ i s r e p r e s e n t e d t h e dependence

Ijr

ⁱⁿ c a s e s when

fa fa

/J,

=0,15 and

P,,+3

^EO (unbroken cur- ve) and ~ : ~ = 0 , 1 5 ~ jn,, ⁽²¹ =O ( d o t t e d c u r v e ) . It i s c l e a r t h a t

ik

t h e f i r s t ca- s e , t h e bunching i s going much q u i c k e r and more e f f e c t i v e l y than i n the f i r s t one.

Dl. It i s shown t h e i n j e c t i o n of a second beam r e s u l t s in t h e beam plasma system becomes i n s t a b l e on t h e f r e q u e n c i e s exce- e d i n g t h e plasma frequency

.

Also t h e second beam a f f e c t on t h e am- p l i t u d e s and spectrum of e x c i t e d waves.

REI"I3r'iEPJCES :

1. V.D.Pedorchenko, Yu.P.L~azalov, k.S.Bnkai, B.N.Rutkevich, 2h.T.F.

(USSR), 65,2225 ( 1973).

2. A .I.Rogashkova, 1L.B. T s e i t l i n , Zh.T.F., ( U S S H ) , X I N , 2514 (1975).

-

-4