Equilibrium phase instability in the double RF system for Landau damping

A longitudinal coupled bunch instability in an electron storage ring was suppressed by the Landau damping in a double rf system composed of a second harmonic rf cavity. The damping became ineffective, however, above a beam current of 30 mA; the beam bunch slipped out of the optimum phase of the total rf voltage for the damping, which accompanied a simultaneous deformation of the total voltage. The unexpected phenomenon of the phase slip is explained by the concept of equilibrium phase instability of the beam bunch based on a rigid bunch model. The phase slip of the bunch was suppressed by introducing a phase feedback loop, resulting in an improvement of the maximum beam current for the damping. Discussions are made on various conditions of the equilibrium phase instability, including another possibility for avoiding the phase slip.