Instability of relativistic electron beam with strong magnetic field LIU ~hijing',H.L. ~ e r & k HE ~ xiantu3 1. Department of Astronomy and Applied Physics, University of Science and Technology of China, Hefei 230026, China; 2. Institute for Fusion Studies, University of Texas at Austin, Austin, TX 78712-1060, USA; 3. Institute for Applied Physics and Computational Mathematics, Beijing 100088, China

Abstract Stability criteria for a weak relativistic beam-plasma interaction in a strong magnetic field are found. Two beam modes occur, w= kfi and w= k s -@. The dispersion equation of electrostatic two-stream is exactly solved with analytical method. Keywords: instability, relativistic electron beam, two-stream, strong magnetic field, dispersion relation. Two-stream instability appears in magnetic confinement devices and laser fusion as well as relativistic electron beam fusion. It plays an important role in ion acceleration and ion compression as well as the ignition of target. In order to make full use of the instability and further control it, it is necessary to understand clearly the relationship between the instability and various parameters and the existent region of the instability. The relativistic two-stream instability has been investigated theoretically in the limit of zero magnetic field1", but there does not appear to be an analysis in the literature for this mode in a strong magnetic field. In this work we analyze the fluid equations for the beam and background electrons and delineate the stability parameters and growth rates. We find that electromagnetic effects are important over a wide range of parameters. We also discuss electrostatic two-stream instability excited by relativistic cold electron beam. Although the dispersion relation was discussed by many a researcher in the past, only a few of its approximative solutions were obtainedr2*'. In this note we also solve exactly the dispersion relation of electrostatic two-stream by using analytical method and calculate numerically the growth rate.

1 The dispersion relation in strong magnetic field Both the beam and background plasma are described in the pressureless fluid limits where the beam speed is taken as vb along the z directed magnetic field B, while the background electron speed is zero. The self field of the beam is assumed to be much less than B, and is neglected in this analysis. Similarly, toroidal effects are neglected. We assume w, q,,<>l . The perturbed equations are the form'). 1) Berk, H. L., Relativistic beam plasma instability in strong magnetic field, Report in Lawrence Livermor Lab., 1972, preprint 1-24.

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I

cat

+ 4 x f?[(ne&, +hb~b)i+&b&'b].

We consider perturbations to be of the form hxp[-i(m - k, z - k,n)]. From the perturbed equations of mass and motion and electromagneticfield we obtain the following dispersion relation:

where

2 Analysis of instability To analyze the stability of eq. (2), we assume a<<%, a.Therefore in eq. (2) we need only keep terms linear in Hence we take A = /(fi2 - w:) . The interesting regimes for stability then

mi.

oi

occur near the resonances where (A) 6 = 0 and (B) fi2 = W:

. For case (A) we can set A

4 and for case

mi

(B) we can set / y2m2= 0 . ( i ) For case (A) f i = 0 . (2) can be approximated as

k: =w; / t 2 +mi l [ y 2 ( ( - v b ) 2 ] + k ~ ~ 2 / (-cZ). t2

(3)

where C =uJk,. If k,: = 0 , eq. (3) is unstable. Thus we have 1-a;

lo2-w; /[y2(u-k2vb)']=0.

(4)

This is a purely electrostatic case. It will be discussed in sec. 3 for cold plasma beam. If we choose a=++&I and +/kz=vb,we then find from (2), that the unstable root is given by

1f k: is sufficiently large, eq. (3) is stable (see fig. 1). In fig. 1 typical curves are shown. For O < t (vb, two possibilities are evident, either the minimum occurs for kl >O (solid curve) or for k: >l. Therefore { m i , * ~We b xcan ~ .write eq. (3) approximately as Chinese Science Bulletin

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NOTES

-

-

- -

-

---

-

--

-

- -

-

k: =a&/ v : +o: ~ [ Y ~ ( < - v ~ ) ~ ]

+ kjc /[2(< - c ) ] . Stability occurs for (a) k , 2

s, if wi < F y2

','. ......,' > vb,1 l

,I

I

I

I I

I

II

I

i

114

;c

-

5

I I y2w: >> w 2p / 4 .

(ii)Forcase(B) 6 = w , , 0 ~< c p , , e q . (2) can be approximated as

Let us choose c& as the solution of eq. (7)when

oi

I

Fig. 1. The curves for p,vs. {.

is completely neglected. We then obtain

, To simplify the results we use the following approximate relations for 4 a$ = k ~ c 2 0 & / ( w ; + k 2 c 2 and@; ) = w k + k2c2 .

They are pure oscillatory solutions. In the limit %< kzvb*c&>>%, we have -kzq,Jk which is an electrostatic oscillation. Instability can be found for k > W: 1 v l . When kv, 1 w, = & ,the maximum growth rate Q is

Now let us investigate the other limit o$<%
When [l+(k2C21upe)]=4,the maximum growth

3112 W I =0,.

(11)

4

If we combine eq. (10)and eq. ( 1 I ) , we find

3 Electrostatic two-stream instability on cold relativistic electron beam Let us set u = v b , w 2B e= y o b =

4nnbe2 1 , y I =--= (1-u2 l ~ ~ ) ' / =~k, .kThus, , from eq. (4) we m Y

have

Eq.(13)can be written as an algebraic equation of fourth order for w u4+ a 1 ~+ a3 2 u 2+ a j u + a 4=o, in which 20

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a, =-2ku,a2 = k 2 u 2 - a , 2- y l ~ 3& , a 3

2 =2kuo,,a4

2 =-k 2 u 2 up.

- ---

(15)

According to the formulae of solving fourth order equationr7',we can obtain

where 2 3 2 A = - - {1 [ a , 2 + y 3l a 2B ,-k 2 u 2 ) 3 +54k 2 u 2 a,ylaB,l 3 2 - 6 & k u ~ , ~ ~ ' ~ ~ ~ , [+(ya: ~ k : , - k 2 u 2 )3 + 27k 2 u 2 m,Yl

2 112 113 DB, 1

.

+ y:a:, - k 2 u 2 )3 + 27k 2 u 2 ~ ,2Y I3@B,2 1112 1113

.

3

1

1 2 2 3 2 2 2 3 2 ~ = - - { [ c u k + ~ : a i , - k u ) +54k u a , y , ~ ~ , ] 3

+ 6&kua,

y:"wB,[(a;

(18)

From the condition (15) satisfied by cubic equation with complex conjugate roots, it can be found that boundary of instability and damping is valid for (ok+y:o:e-k

2 3 2 u ) +27k 2 u 2 a,ylw,=O.

2 2 3

This is a cubic equation for kZu2. From Cadar formula we have found

When y:wi, 1 a&<< 1, we have Chinese Science Bulletin

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(19)

NOTES

obvious,y,

) :1

ku

-5 Ope

C

In the case of collision against each other for n&= n,, we have

For convenience's sake, write

Thus 3

[ a = ( l l y , u , -u,

2 3

Hence we have found from (16)-(18),

2 3

+549 y,u,,

(22). (23) that

(25) Since electron energy ~ = r n c ,~ ~l ~ , =,m c ~ 1 ~ =11/E 0.5 [MeV]. The values of cyl+ and CO,/I+ can then be calculated from (16), (17), (23)-(25) as E, u2 are given Fig. 2. The curves for (see figs. (2-4)). varies). 22

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a+ vs. ul (as m.6

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Fig. 3. The curves for ~ 1 %vs. u , (as E=l.O MeV, u2 varies).

u2 =0.05

E = 0.6 MeV

Fig. 4. The curves for ~ 1 %vs. ul (as u2=O.OS, E varies).

4 Conclusion and discussion Stability criteria for a weak relativistic beam-plasma interaction in a strong magnetic field are found. l b o beam modes occur, w=k& and w*k,c- &. The former mode is stabilized if the k, is sufficiently large. For electrostatic two-beam mode we have obtained analytic expressive forms (24) and (25). These solutions are general. The result of numerical calculation is expressed as figs. 2-4. The maximum of growth rate w , shifts to the region of ku>+, the growth rate and the region of instability became larger with increasing density ratio nBe/ne,also became less with increasing energy, and effect of relativity can suppress electrostatic two beam instability, too. It can be seen that the higher the energy of relativistic electron beam, the larger the application range of approximation of weak stream. References 1.

2. 3. 4. 5.

6. 7.

Bludman. S. A., Watson, K. M., Rosenbluth, M.N., Statistical mechanics of relativistic streams I , 11, Phys. Fluids, 1960. 3(5): 74 1. Akhiezer. A. I., Akhiezer, I. A., Polovin, R. V. et al., Plasma Electrodynamics ( I ), New York: Pergamon Press Ltd, 1 9 5 , Chapter 6,339. Breizman, B. N., Collective interaction of relativistic electron beams with plasmas, in Reviews of Plasma Phys. (ed.Kadomtsev, B. B.), New York: Consultants Bureau, 1990, 15: 61. Buti, B., Relativistic effects on plasma oscillations and two-stream instability I ,1I, Phys. Fluids, 1963,6(1): 89. Thode, L. E., Sudan, R.N., -0-stream instability heating of plasma by relativistic electron beams, Phys. Rev. Lett., 1973, 30(16): 732. Ferch, R.L., Sudan, R.N., Linear two-stream instability of w k relativistic electron beams, Plasma Phys., 1975, 17: 905. Hua Luogeng, Introduction for Advanced Mathematics (in Chinese), Beijing: Science Press, 1974, l(1): 34. (Received October 8, 1998)

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January 2000