Calculation of Transverse Coupled Bunch Instabilities in Electron Storage Rings Driven By Quadrupole Higher Order Modes

This paper presents a formula that estimates the growth rate of a transverse coupled bunch instability driven by quadrupole higher order modes (HOMs) in electron storage rings. Thus far, quadrupole HOMs are usually ignored in HOM driven instability studies for electron storage rings due to their weak nature compared to the lower orders. However, they may become relevant when high gradient SC multi-cell cavities with their potentially strong impedance spectrum are operated at high currents in a third generation or future synchrotron light source. An example is BESSY VSR, a scheme where 1.7 ps and 15 ps long bunches (rms) can be stored simultaneously in the BESSY II storage ring [1]. With the presented formula, instability thresholds are discussed for a recent BESSY VSR cavity model and different beam parameters. INTRODUCTION Coupled bunch instabilities (CBIs) driven by higher order modes (HOMs) of RF cavities or other narrow-band impedances are a classic type of instability encountered in many storage rings. An extended study of CBIs relating to BESSY VSR, the upgrade scheme of BESSY II to obtain short and long bunches simultaneously [1,2], has recently been published [3]. This paper presents findings related to quadrupole HOMs. The consequence of a CBI is either beam loss, usually in the transverse plane, or a saturation of the beam oscillation at very large amplitudes if it is occurring in the longitudinal plane. In both cases, the instability must be avoided to ensure the high beam quality promised by all low emittance storage rings. Common methods to suppress CBIs include the usage of an active bunch-by-bunch feedback (BBFB) in each plane that tracks the dipole motion, i.e., center of mass motion, of the beam and applies kicks such that the oscillation amplitude is reduced. Typically, studies of HOM driven CBIs consider only two special cases, namely the dipole motion in the longitudinal or transverse plane caused by the interaction with monopole or dipole HOMs respectively. Formulas to estimate the growth rate of these instabilities based on the machine parameters and the impedance spectrum are used frequently in literature, see for example [3] and references therein. If the growth rates are compared to a damping rate, e.g. the damping rate ∗ Work supported by German Bundesministerium für Bildung und Forschung, partly under contract 05K13PEB, and Land Berlin. † martin.ruprecht@helmholtz-berlin.de of the BBFB, a threshold impedance is found. The threshold impedance for the longitudinal and transverse dipole CBIs driven by monopole and dipole HOMs is given by [4] Z ‖ 0,th(ω, τ −1 d ) = τ−1 d ωs ωα 4πE/e ωrevI , (1) Z⊥ 1,th(τ −1 d ) = τ−1 d β 4πE/e ωrevI , (2) respectively, where the first index of Z represents the order m of the impedance and ω the angular frequency at which the impedance is sampled, ωrev the angular revolution frequency,ωs the angular synchrotron frequency, τ−1 d the damping rate, α the momentum compaction factor, E the beam energy, e the elementary charge, β the betatron function at the position of the cavity. As an example, Fig. 1 shows the impedance of the 1.5GHz cavity model of the BESSY VSR project [5] compared to the threshold impedance based on the BESSY II machine parameters and the present BBFB damping performance [1–3]. 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 f/GHz 10-6 10-4 10-2 100 102 104 106