Dispersion in the Presence of Strong ~ansverse Wakefields*

To minimize emittance growth in a long linac, it is necessary to control the wakefields by correcting the beam orbit excursions. In addition, the particle energy is made to vary along the length of the bunch to introduce a damping, -known as the BNS damping, to the beam break-up efiect. In this paper, we use a two-particle model to examine the relative magnitudes of the various orbit and dispersion functions involved. The results are applied to calculate the etiect of . a closed orbit bump and a misaligned structure. It is shown that wake-induced dispersion is an important contribution to the beam dynamics in long linacs with -strong wakefields like SLC. Presented at the 17th IEEE Particle Accelerator Conference (PAC 97): Accelerator Science, Technology and App!icatiom, Vancouver, B. C., Canada, May 12-16,1997 *Work supported by Depwtment of Energy contract DE-AC03-76SFO0515. DISPERSION IN THE PRESENCE OF STRONG TRANSVERSE WAKEFIELDS* Ralph Assmann and Alex Chao Stanford Linear Accelerator center, Stanford University, Stanford, CA 94309 Abstract To minimize emittance growth in a long linac, it is necessary to control the wakefields by correcting the beam orbit excursions. In addition, the particle energy is made to vary along the length of the bunch to introduce a damping, known as the BNS damping, to the beam break-up effect. In this paper, we use a two-particle model to examine the relative magnitudes of the various orbit and dispersion functions involved. The results are applied to calculate the effect of a closed orbit bump and a misaligned structure. It is shown that wake-induced dispersion is an important contribution to the beam dynamics in long linacs with strong wakefields like SLC.To minimize emittance growth in a long linac, it is necessary to control the wakefields by correcting the beam orbit excursions. In addition, the particle energy is made to vary along the length of the bunch to introduce a damping, known as the BNS damping, to the beam break-up effect. In this paper, we use a two-particle model to examine the relative magnitudes of the various orbit and dispersion functions involved. The results are applied to calculate the effect of a closed orbit bump and a misaligned structure. It is shown that wake-induced dispersion is an important contribution to the beam dynamics in long linacs with strong wakefields like SLC. 1 WAKE ~DUCED DISPERSION (TWO-PARTICLE MODEL) Consider a linac with uniform betatron focusing and no acceleration. Introduce an orbit kick 6 ats = O. The betatron equation of motion for a particle with relative energy error d is k; x“(s) + I:dd(s)’ =x(s) = — (1) with the solution 9 x(s) = k~m sin(~). (2) When k)vsd <<1, one may expand (2) in 6, i.e. z(s) = Xa($) + q(s)ti + 0(62), (3) with 9 XO(S) = — sink13s k,~ and the dispersion function q(s) = – ~ (s cos k13s + ~ sin k6s). When kps >> 1, the dispersion effect can c~early be important. When kpsd << 1 is not satisfied, we will have to use Eq.(2) instead of Eq.(3). We next consider a two-particle model for the kicked beam. The motion of the leading macroparticle (considered to be on-momentum) of the beam is given by x = X.(s). Let N/2 be-the number of electrons in the leading and the trailing macroparticle, ~ be the design energy Lorentz factor, W1 be the wake function per cavity period, and L be the cavity period length. Let y(s) designate Ihe orbit deviation *Worksupportedby the Departmentof Energy,contractDE-AC0376SFO0515. of the trailing macroparticle. We have y“(s) + 1 !d~(s) ~y(s) = — NTO W1 — 1 sin k,ds, (4) 27L(1 + d) kp where To is the classical electron radius. In Eq.(5), we have assumed the leading and the trailing macroparticle have the same design betatron frequency. This is the case when there is no BNS damping. The case with BNS damping is to be treated later. The solution to Eq.(5) is To 2 y(s) = 3(s) – ——. kPLo kpb ( kps sin kps – -sin m ) (5) where we have defined a dimensionless parameter ~ = _NroWILo 4yLkP . (6) The first term on the right hand side of Eq.(6) is the direct -response of the trailing macroparticle to the orbital kick and is the same as Eq.(2). The second term is the driven response to the wakefield. When k~sd <<1, we can expand (6) in b to obtain y(s) = ye(s) + [q(s) +((s)]6 + o(ti2) Ye(s) = ~o(s) – *(s coskfis – ~ sink,fis)