Landau damping via the harmonic sextupole

Abstract

Multibunch instabilities of a storage ring electron beam occur due to coherent particle oscillations generated through a bunch to bunch coupling via the impedances, deteriorating the beam quality. One cure for multibunch instabilities is Landau damping, i.e., introducing a spread in the oscillation frequencies among the particles of the individual bunches in order to destroy the coherence of the coupled multibunch oscillation. Measurements at ELETTRA have shown that the harmonic sextupole provides Landau damping capable of suppressing transverse multibunch instabilities. The damping is induced by the nonlinear tune spread with amplitude among the electrons within the individual bunches.

Review of collective beam instabilities in electron-positron storage rings

“…The required spread of frequencies in the beam can be ensured by means of nonlinear elements of the magnetic lattice, namely, by octupoles and sextupoles. Octupoles involve an amplitude dependent shift of a betatron frequency as the first order effect, while sex tupoles (whose main effect on the beam is a frequency shift depending on a longitudinal momentum (for rel ativistic particles, on energy)) also introduce an amplitude dependent shift of a betatron frequency, which, as the investigations performed at an Elettra SR source of a third generation have shown [75]), may turn out to be sufficient to ensure the required spread of the frequencies. Since a multibunch instability can be considered as a set of two types of a collective motion of particles, i.e., oscillation of each bunch as a whole and oscillation of the particles inside the bunch, a principle question arises as to whether the reason for the multibunch instability suppression is a spread of frequencies between the bunches or inside each bunch.…”

“…Thus, the experimental study of transverse multi bunch instability of the beam in an Elettra storage ring [75] showed that instability can be efficiently sup pressed by decoherence of oscillations of the beam particles that is introduced by the family of harmonic sextupoles. The results of the study clarified that atten uation is caused by the spread of the frequencies of oscillations of the particles inside the bunch, and it may be efficient both in the cases of multibunch and single bunch instabilities.…”

“…For N e → 0 the frequency of mode is Ω → ω β + lω s . The matrix has the following form (75) where ω ξ = ξω 0 /α, g lk are the functions which charac terize the oscillation modes and, in the case of Gauss ian beam, they read as (76) The analysis of the shift of frequencies and the increments of the coupled modes of oscillations depending on the electron cloud density and chroma ticity, with and without the feedback, is given in [57]. Two facilities were considered: KEKB LER and CERN SPS.…”

“…Under the resonance conditions all bunches perform coherent oscillations with a certain phase advance between bunches. A transverse multibunch instability was observed and studied at the following facilities: PEP II (United States) [71], APS (United States) [72], DAFNE (Italy) [73], ELETTRA (Italy) [74,75], ESRF (France) [76], and KEKB (Japan) [77].…”

The nonlinear effects of fringe fields in HLS

“…At HLS, there haven't other reliable beam stabilizing measures, such as longitudinal and transverse feedback systems, non-uniform filling mode, except for Landau damping, which is coming from transverse tune spread and becomes a main stabilizing mechanism at storage ring. Usually, the second order effects of sextupoles, which used to compensate chromaticities or to enlarge dynamic aperture, would result in tune shift with amplitude in transverse planes [2,3] :…”