Scaling law of coherent synchrotron radiation in a rectangular chamber

Abstract

We study the threshold of longitudinal instability driven by coherent synchrotron radiation in a rectangular chamber in electron storage rings. Starting with Maxwell’s equation, we first derive a point-charge wakefield for longitudinal motion. Then we use the wake, along with the Vlasov-Fokker-Planck equation, to show that the threshold can be described by a simple scaling law of ξ=ξ^{th}(χ,A,β[over ^]), where ξ is a dimensionless current, χ is the shielding parameter, A is the aspect ratio of the vacuum chamber, and β[over ^] is the damping rate relative to the rate of synchrotron oscillation. We further investigate the threshold with simulations for various values of aspect ratios. In particular, for a square chamber, we find that ξ^{th}≈0.25, which is a factor 2 lower than the threshold in free space.