Exact solution of multi-bunch instabilities for ultra-relativistic constant energy bunches in particle accelerators

Abstract

In particle accelerators the multi-bunch beams of charged particles excite electromagnetic wakefields in the accelerating cavities and other beam-line components. These wakefields may cause instability in the longitudinal and transverse motion of these bunches leading to the beam loss. Thus, the analysis of such multi-bunch effects is critical in any accelerator design. Here, we present a new analytical method to exactly describe the multi-bunch transverse instabilities for ultra-relativistic electron bunches coasting at a constant energy through a uniform beam-focusing system. We discuss two cases: same bunch energy for all bunches and different energy from bunch to bunch. For the first case the solution is based on a special class of polynomials, while for the second case the solution is a finite sum of sinusoidal functions. We demonstrate the use of this approach with examples and compare the results of a numerical differential equation solver against our rigorous results. Our method could also be applied to the analysis of beam breakup within a single bunch. This approach can be extended beyond accelerator physics to study other many-body systems where an body is affected by only the preceding bodies.