Longitudinal Stability of Multiturn ERL with Split Accelerating Structure

Some modern projects of the new generation light sources use the conception of multipass energy recovery linac with split (CEBAF-like) accelerating structures [1 4]. One of the advantages of these light sources is the possibility to obtain a small bunch length. To help reduce it, the longitudinal dispersion should be non-zero in some arcs of the accelerator. However small deviations in voltages of the accelerating structures can be enhanced by induced fields from circulating bunches due to the dependence of the flight time on the energy deviation and the high quality factor of the superconducting radiofrequency cavities. Therefore, instabilities caused by interaction of electron bunches and fundamental modes of the cavities can take place. The corresponding stability conditions are discussed in this paper. Numerical simulations were performed for two projects – MARS [4] and FSF [3]. INTRODUCTION The proposed scheme of the fourth generation light source project based on multiturn accelerator-recuperator with two accelerating structures is shown in Fig. 1. Figure 1: Scheme of MARS ERL with two linacs. Electrons from injector 1 pass through the preliminary accelerating structure 2 to the main accelerator. Obtaining necessary energy in the main linacs 3 electrons are used in undulators 4. After deceleration, electrons are dropped to the dump 5. There are four electron beams in each main linac simultaneously. Each beam induces large voltage in the linac, but the sum is not so large. If the phases of the beams vary, the accelerating voltages also change, and initially small phase deviation may increase due to the dependence of flight times through arcs on the particle energy. This longitudinal instability is considered in our paper. THEORY To simplify the picture, consider each linac as one RF cavity. Taking the effective voltage on the linac with number α in the form ) Re( t i e U    (ω is the frequency of the RF generator), one obtains: ), ( 1 2           g b I I U Q i dt dU     (1) where         ) 2 / 1 ( / 1 Q C L    is the resonant frequency,     C L R Q  >>1 is the loaded quality of the cavity, C L Q R       / and  R are the characteristic and the loaded shunt impedances for the fundamental (TM010) mode,  g I and  b I are the complex amplitudes of the beam and (reduced to the gap) generator currents correspondingly. We are interested in the case of constant  g I . The beam currents  b I depend on all Uα due to the phase motion. Linearization of Eq. (1) near the stationary solution gives:            U Q i dt U d 1 2 (2)                          U U I U U I b b Im Im Re Re Considering the exponential solutions   2 exp t  of system of linear differential equations Eq. (4), one can find the stability conditions. Indeed, the system Eq. (2) corresponds to the system of the linear homogeneous equations U M U    δ with the consistency condition 0   E M  . Then Re(λ) < 0 for all roots of the last equation (i. e., eigenvalues of the matrix M) is the stability condition. The explicit expression for M is