Investigation of Resistive Wall Instability in the 7-GeV APS Storage Ring

The Advanced Photon Source (APS) storage ring is a 7-GeV light source with 40 straight sections. Intense x-ray beams will be delivered by 34 insertion devices installed in these straight sections. The vacuum chamber for the insertion devices has an elliptical cross section with the gap equal to 8 mm. With this narrow gap, we estimate that the transverse impedance of the ring at the revolution frequency could be as high as 36 MÄ from the resistive wall. By increasing the (unnormalized) chromaticity to 7, we cure the head-tail modes of order up to m=1 for all 60 coupled bunch mode patterns around the ring. Tracking results show that the increased sextupole strength resulting from a higher chromaticity does not significantly reduce the dynamic aperture. Since increased chromaticity alone cannot cure all the head-tail modes, the APS storage ring will have a feedback system to damp the rigid-bunch modes. Introduction Resistive wall impedance can cause the coupled bunch instability due to the peak near the origin (long range wakefield or multi-turn effects) as well as the higher-order head-tail modes via the broad-band tail (short range wakefield or singleturn effects). Since the growth rate from the resistive wall instability is in general slow, the strategy is to damp the fastest growing mode of the coupled bunch oscillation by adjusting the chromaticity slightly above zero, causing the unstable head-tail modes to become stabilized by the radiation damping and/or Landau damping. However, we found that this is not the case for the APS storage ring. Resistive Wall Impedance R. Gluckstern, J. Zeitzs and B. Zotter [1] have derived expressions for the longitudinal and transverse resistive wall coupling impedance for a beam pipe of arbitrary cross section in the ultra-relativistic limit. Explicit results for the transverse impedance for the beam pipe of elliptic cross section with the major axis a and the minor axis b may be written Zx,y(ω) = (1+ j) Z0δL 2πb3 Fx,y(q) ≡ Z⊥,circular (b, ω)Fx,y(q), (1) where Z⊥,circular (b, ω) is the transverse impedance for the cylindrical beam pipe of radius b and Fx,y(q) is the form factor expressed in terms of “nome” q = (a−b)/(a+b). The subscripts x and y denotes the horizontal and vertical impedance, respectively. Denoting Z y as Z⊥ and using the fact that the vertical form factor, Fy(q) is bounded by 0.8 and 1.0 for the entire range of q , we may approximate Eq. (1) as Z⊥(ω) ' Z⊥,circular (b, ω). (2) ∗Work supported by U.S. Department of Energy, Office of Basic Energy Sciences under Contract No. W-31-109-ENG-38. Then, Eq. (2) may be rewritten as Z⊥(ω) = (sign(ω)+ j)Z⊥(ω0) √ ω0