Analytic formulas for the D -mode Robinson instability

The passive superconducting harmonic cavity (PSHC) scheme is adopted by several existing and future synchrotron light source storage rings, as it has a relatively smaller R/Q and a relatively larger quality factor (Q), which can effectively reduce the beam-loading effect and suppress the mode-one instability. By using a minimum search algorithm to solve the mode-zero Robinson instability equation of uniformly filled rigid bunches, we have revealed that the fundamental mode of PSHC with a large loaded Q possibly triggers the D-mode Robinson instability [T. He , Mode-zero Robinson instability in the presence of passive superconducting harmonic cavities, ]. This instability is a mode-zero coupled bunch instability, with an oscillation frequency close to the PSHC detuning (D-mode). Uniquely, it is anti-damped by the radiation damping effect. In this paper, we derive analytical formulas for the frequency and growth rate of the D-mode Robinson instability by taking several appropriate approximations. These formulas provide crucial insights for analyzing and understanding the D-mode Robinson instability. Published by the American Physical Society 2024