Longitudinal BBU Threshold Current in Recirculating Linacs

ERL-06-04 The total current that can be accelerated in an Energy Recovery Linear Accelerator (ERL) can be limited by the longitudinal recirculating beam-breakup instability (BBU), as well as the transverse BBU instability. In this report we discuss a simple model of the longitudinal BBU instability that can be solved analytically. More complicated cases are analyzed by means of numerical simulation. It should be noted that the discussion in this report applies to all recirculation linacs and the ERL is a special case with recirculation phase . The theory and simulation are compared and they agree with each other very well in the simple model case. Similar to the transverse case, the in uence of the higher order modes (HOM) frequency spread among dierent RF cavities is also discussed. Moreover a theory of the longitudinal HOM power growth is presented and the analytical result agrees well with the simulation. 1 Longitudinal Beam Breakup Instability When an electron bunch passes through an RF cavity, it can excite both transverse and longitudinal HOMs. Previous papers have described how the recirculation of electron bunches can cause the exponential growth of tansverse HOM power leading to transverse BBU instability [1, 2, 3, 4]. Similar to the transverse case, longitudinal HOMs excite