Linear theory of microwave instability in electron storage rings

Abstract

The well-known Haissinski distribution provides a stable equilibrium of longitudinal beam distribution in electron storage rings below a threshold current. Yet, how to accurately determine this threshold, above which the Haissinski distribution becomes unstable, is not firmly established in theory. In this paper, we will show how to apply the Laguerre polynomials in an analysis of this stability that are associated with the potential-well distortion. Our approach provides an alternative to the discretization method proposed by Oide and Yokoya. Moreover, it reestablishes an essential connection to the theory of mode coupling originated by Sacherer. Our new and self-consistent method is applied to study the microwave instability driven by commonly known impedances, including coherent synchrotron radiation in free space.