The 9th Symposium on Acceleraior &ience and Technology, Tsukuba, Japan 1993

Transvers e two-stream Instability In a matched plasma channel
David H. Whlttum
National Laboratory for High Energy Physics (KEK),
1-1 Oho, Tsukuba, lbaraki, 305, Japan

Abstract
A relativistic electron beam magnetically selffocused in a plasma is subject to a transverse two streamor "hose• instability. Linear evolution is described in termsof a tune distribution characterizing the beam, and an
effective transverse impedance determined by the beamand the plasma profiles. This model is compared to cloudin-cell simulations of three-dimensional transport of abeam with a Bennett profile, through a matched plasmachannel. In the limit of large skin-depth this instability
appears to be the primary limitation on stable beamtransport.

tis time, z is axial displacement and cis the speed of light.
The beam propagates through a much denser plasma
maintaining quasineutrality in equilibrium, provided thebeam is several plasma periods in duration. Negligible
plasma return current flows through the beam volume
provided the plasma skin-depth is larger than the Bennett
waist. ion-motion and radiative effects are neglected. Theequilibrium plasma is assumed stationary in -r, created
rapidly by the beam head. We assume for simplicity thatthe plasma profile is matched to the beam.
To the Bennett equilibrium consider a rigid beamdisplacementY(z,-r). Combining the Vlasov equation andthe linearized cold fluid equations one can show thatmomentum conservation for the beam takes the form

I. INTRODUCTION
In recent years the roles of plasma in accelerators
have been the subject of many works (1-5] emphasizingthat plasmas provide strong coupling, for focusing and foracceleration making use of relativistic electron beams. It isless widely appreciated that this strong coupling extends tostrong deflection and break-up of the beam. In otherarenas, plasma-coupled instabilities have motivatedvigorous studies over many years, all concentrating on theresistive hose instability [6-8], or the filamentation
instability (9,10]. In this literature there is scant reference tothe transverse two-stream instability (11]. Here we willshow that the electrostatic mode of beam break-up istypically more virulent than any other, in the limit of largeskin-depth, the limit of primary interest for futureaccelerator applications.

(1)

where a Laplace transform (indicated by the tilde) has
been made in -r, with p the Laplace transform variable. The"slosh" wavenumber ks is ksfkp=3·112 for the Bennett
equilibrium, with kp the wavenumber for small amplitudebetatron oscillations. The normalized
impedance Z may be expressed as
2k, z( p) =-2 1e r. c 2 ( ()rTr( rvr)

transverse)'(2)

where re is the classical electron radius, r is the Lorentz
factor for the beam and the brackets denote an averageover the unperturbed beam. The pinch potential vr isdetermined from the solution of the elliptic equation in r,

II. ANALYSISWe consider a relativistic electron beam with a
Bennett-profile [12] charge density Pb, a function of theradial coordinate rand the beam coordinate -r-t-z/c, where-413-(fr Jpb
J(jr1 ()
- =Jr- r eJr- - r2r Jr(3)according to yt(r,p)= QF( r,oo)- QF( r,p), The cold
plasma dielectric function is just £•1 +CtJ,,2(r}lp(p+v) with CtJ,
the local angular plasma frequency and v aphenomenological plasma electron collision rate.

zIn general it is not possible to solve foranalytically; numerical solution is however straightforward
and is depicted in Fig. 1 for the Bennett-profile plasma ofinterest here, with V=O. The wake obtained with an Inverse
fast Fourier transform is shown in Fig. 2. This result is wellfit with a single-mode Lorentzian

2II"\\ICtJz

Z( p)...

r.
(t)~ + P(P + Vr.)(4)
IN owith resonant frequency CtJL -0.71 CtJe and damping rate
VL -0.34CtJe. The corresponding wakefield (the inverse...

- 1

Laplace transform of Z) is overlayed in Fig. 2.
With this result it is straightforward to show that fork 5 Z<<CtJ 9 7:, the convecting peak in amplitude varies

IIII

L-------L-~-~----~------~------~-2- 2

12FIG. 1. The normalized transverse impedance of aBennett plasma column, for a Bennett beam, is well-fit by asingle-mode Lorentzian with Q-2.

asymptotically as Y- exp(z!Lg) where k5 L9 -1, i.e., growth
proceeds rapidly, on the scale of the betatron period.
These results apply for short-range propagation,
kpz-0(1). Over a long range, growth is diminished byphase-mixing in beam electron motion. This effect can bemodelled following Lee's work on the resistive-hoseproblem[6]. The beam centroid is represented as anaverage, Y = Jd a g( a)Y.,. The "mass ,distribution"

0.6

g=6a (1-a)
is normalized to unit integral, thedimensionless parameter a lies in the range [0, 1], and thecomponents satisfy

0.3~

8~'/~0(5)'I1\.

-exact-0.3--single

mode-0.6
01 04 0The corresponding dispersion relation is readily solved toestablish that the saturation length L5 , scales according tokpL:/CtJ!:r-3.4, with amplitude at saturation Y- exp(Pr) ,where T-0.9CtJL.
Ill. SIMULATION

FIG. 2. The wakefield corresponding to the impedance ofFig. 1, overlayed with the single-mode fit.

To check this model we make comparison with acloud-in-cell (CIC) simulation. This simulation advances
the plasma variables (transverse position andnonrelativistic momentum) in the beam coordinate 1', with aleap-frog algorithm governed by the electrostatic potential.

-414-

The corresponding beam variables are advanced in z by aleap-frog algorithm, governed by the pinch potential. The
neutralizing ion background is fixed. This formulation is

gradient considered here. Unfortunately this "cure• will
also damp the often desirable longitudinal wake.

consistent with (and limited to) large plasma skin-depth, anultra-relativistic beam, and negligible radiative effects.
The saturation amplitude from the CIC simulation isACKNOWLEDGMENTS

shown in Fig. 3, overlayed with the model results, obtainedby solving Eq. (5) in the time domain. A least-squares fit tothe PIC results gives r-mL. For a pulse 1.5 plasma periods

This study has been informed and motivated bycollaborative efforts on related problems with Glenn Joyce,
Martin Lampe, William M. Sharp, Steven P. Slinker, andin length, saturation occurs after growth by a factor of1x102, for a uniform initial offset.

SimonS. Yu.
REFERENCES[1] P. Chen, Phys. Rev. A 45,3398 {1992).

//-CIC[2] P. Chen, et at., Phys. Rev. Lett. 54, 693 (1985).

//--MAC/[3] S. B. Swanekamp, et at., Phys. Fluids B 4, 1332 (1992).

/~/

1 o1

(4] J. B. Rosenzweig, eta/., ibid., 2, 1376 (1990).

///[5] H. Nakanishi, et at., Phys. Rev. Lett 66, 1870(1991).

/y[6] E. P. Lee, Phys. Fluids 21, 1327 (1978).

///[7] E. J. Lauer, et at., ibid., 21, 1344 (1978).

1 o0024681 0(J)e1:

FIG. 3. Results for saturation amplitude from thenumerical solution of the macroparticle model, the CIC
simulation.

[8] M. Lampe, et at., ibid., 27, 2921 (1984).
[9] E. S. Weibel, Phys. Rev. Lett 2, 83 (1959); Ronald C.
Davidson, Physics of Nonneutral Plasmas (AddisonWesley, Redwood City, 1990).

IV. DISCUSSION[1 OJ R. Keinigs and M. E. Jones Phys. Fluids 30, 252(1987); J. J. Su, et at., IEEE Trans Plasma Sci., PS-15,
192 (1987).

Previous analyses of transverse stability have been
performed in the limit of short plasma skin-depth where theeffect of plasma return current is pervasive. Typically the[11] J. D. Lawson, The Physics of Charged-Particle Beams,(Clarendon Press, Oxford, 1977).

plasma was assumed highly collisional, due to a lowionization fraction. In this limit, resistive hose growth has
been a serious concern, with growth length scaling as
k5 L9 --rolr, and -ro the diffusion time-scale. For large skin-(1955).

[12] W. H. Bennett, Phys. Rev. 45, 890 (1934); 98, 1584

depth however, both resistive-hose and filamentation areminor compared to the strong electrostatic plasma
resonance. Control of this beam beak-up mode favors
damping of the wake, for example, with the plasma-415-