Microwave Instability across the Transition Energy

It is well known that during the acceleration of hadrons in a storage ring, the beam always goes above the microwave instability threshold near the transition energy ¿t. The reason is that the longitudinal revolution frequency spread of the beam which otherwise provides Landau damping vanishes at the transition energy. The amount of the beam dilution near the transition energy is determined by ¿th, the length of time when the beam stays unstable, and the growth rate of the instability. It is pointed out in this paper that ¿th is proportional to the fourth power of ¿t, and thus the choice of a large ¿t is not desirable from this point of view. An analysis is also given of the microwave instability induced beam dilution for the proposed Relativistic Heavy Ion Collider at BNL.