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Rf Focusing Effects and Multi-bunch Beam Breakupin Superconducting Linear CollidersJ. Rosenzweig, S. Hartman and J. StevensUCLA Department of Physics, 405 Hilgard Ave., Los Angeles, CA 90024

AbstractA high gradient standing wave linearaccelerator provides axisymmetric transversefocusing due to the presence of strong alternatinggradient transverse electromagnetic fields arisingfrom the backward rf wave. This effect is second

order in both the field amplitude and in y-l, so itis of importance only for high gradient, relativelylow energy beams. The purpose of the presentanalysis is to examine the effect of this focusingon multi-bunch beam breakup in a superconduc-ting linear collider, which has both a high acceler-ating gradient and long bunch train. As aninteresting test case, we discuss the beam breakupproblem in the TESLA test bed at DESY.

I. RF FOCUSINGThe effect of alternating gradienttransverse x-f ields is to provide net axisymmetric(monopole) focusing to a beam accelerating in astanding wave electron linear accelerator. Thefocusing strength associated with this effect is, in

the smooth approximation,I- = 72K=l eEcf2 pm2 1 1where E,, is the average accelerating gradientassociated with the resonant component of thewave[ I]. This focusing effect has the same form asa solenoidal magnetic field, of strengthB, = Eti /&z. Note that for a large averageaccelerating gradient and a low (by the standardof the linear collider final energy) beam that thiscan be a very strong effect. For TESLA designs,an average accelerating gradient of 20 MeV/m isoften assumed, which yields an effective smoothfocusing with equivalent beta-function ofP,,(cm) = 3.5~. At the TESLA test bed at DESY,the beam will be injected at low energy(approximately 10 MeV), and thus the initial beta-function will be 0.7 m, which is much strongerfocusing than the external quadrupole lensesprovide. In addition, if the electron beam atTESLA itself is derived directly from an rf gun [2],

0-7803-1203-l/93$03.00 1993 EEE

the rf focusing will be dominant in the low energy(less than the damping ring energy) section of thelinac. This focusing term can be incorporatedinto a paraxial ray equation, which has thefollowing simple form in the limit that y >> 1,X+3/y)zX =()4\Y)

where X = -x/f i is the reduced transverseposition (the derivation of the general form of thisequation, and the notation, are due to Lawson[3],and y = eEf / mc2 is the average rate of changeof the beam energy in units of rest energy.In addition, at the entrance/exit of thecavities, the unmatched iris fields is are equivalentto focusing/defocusing lens of focal lengthf = 2E, / E,=20 cm[4].

These effects can be incorporated into asingle matrix which describes the passage of aparticle through a cavity,cos(a) -JZsin(a) &$sin(a)

of 13 y .-=Ta) E[cos(a)+JZsin(a)] 1

-Llnwhered and yiYicr,s the normalizedbeam energy at the beginning (end) of the cavity.Using this matrix, plus the linear drift andquadrupole matrices in the intercavity regions ofthe linac, one can construct a full linear transportmatrix for the linac.II. MULTI-BUNCH WAKE FIELDS.

As a bunch which is a member of multi-bunch train passes through a cavity off-axis, itadds a wake-field contribution to the lowest ordertransverse modes, the dipoles modes. Since thevertical emittance and beam size are much smallerthan the horizontal in a linear collider design, weconcentrate only on vertical motion, as it will be amore sensitive measure of a wake-field instability.The total vertical kick received at the N-th cavity

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While this is not quite the case here, the behaviorcan be seen to be qualitatively the same.

I no rf focusing. . . . rf focusing I-0.5; 20 40 60 80 100

bunch numberFigure 2: Vertical position as a function of bunch trainnumber, with and without rf focusing for TESLA test bed-like case,with the initial bunch offset subtracted.

It is also clear that the final lock-on offsetseen in Fig. 2 is slightly larger for the case whichincludes the rf focusing kicks. In the daisy chainmodel the oscillation amplitude is proportional tothe average beta-function. Thus a likelyexplanation for the calculated amplitudes in Fig. 2is that total amplitude is made smaller by thepresence of the additional rf focusing, an effectwhich cannot, however, overcome the addedgrowth of the instability due to the presence of rfkicks, which provide an additional inhomogeneousdriving term in the equations of motion. In bothcases, however, the instability amplitude is arelatively small fraction of the bunch heightoy = 200 pm. This fraction, as stated above,tends to decrease with additional acceleration in aTESLA-type machine, especially in our case,where the beta-function remains constant.

IV. CONCLUSIONSThe effect of alternating gradient rf focus-ing has a large effect on the focusing of electronsat low energy in a standing wave linac. Thisundoubtedly will have some impact on the wayone implements external focusing and trajectorycorrection in an SRF linear collider.On the other hand, we have found noserious deleterious effects of including rf focusingin a multi-bunch BBU in a long tnachine. In a lowenergy machine such as the DESY test-bed,

however, the additional effect of the rf kicksshould be observable.The idea behind pursuing this study wasthat the rf focusing kicks would contribute aforcing of the motion that is correlated to thewake-fields, since both the kick received by thebunch and the wake-field excited by the bunch areproportional to the bunch offset from the cavitycenter. The bunches affected by the wake-field,however, do not in general execute subsequentmotion which is correlated to the cavity offsets,except to the extent that a dipole mode frequencyis commensurate with the bunch spacingfrequency. This special case is to be avoided, andin fact the most desirable choice of dipole modefrequency places the zero crossings of the dipolemodes at the following bunches.[7] Thus themultibunch BBU is not drastically exacerbated bythe presence of rf kicks in cases where the bunchtrain would be otherwise stable, as can be seen inFigure 2.

REFERENCES1. S. Hartman and J. Rosenzweig, Phys. Rev. E47, 2031 (1993).2. J. Rosenzweig, E. Colby, G. Jackson, and T.Nicol, these proceedings (abstract Ma9).3. J. D. Lawson, The Physics of Charged ParticleBeams (Oxford University Press, Oxford, 1977).4. R. Miller and R. Helm, in Linear Accelerators,ed. P.M. Lapostolle and A.L. Septier, 115 (North-Holland, Amsterdam, 1970).5. A. Mosnier, internal note DAPNIA-SEA-91-16(Saclay, 1991).6. C. Adolphsen, SLAC-PUB-5942 (Oct. 1992), tobe published in the Proc of the XVth Int. Conf. onHigh Energy Accelerators.7. K.A. Thompson and R.D. Ruth, Phys. Rev. D,41,41 (1990).

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