Abstract
Head-tail modes in bunches with space charge are studied using particle tracking simulations. The eigenfrequencies and eigenfunctions of transverse coherent oscillations in a Gaussian bunch are determined and compared with theories. A model for an airbag distribution in a barrier potential gives good predictions for the head-tail spectrum and for eigenfunctions in bunches with space charge. Using numerical simulations, space-charge induced Landau damping in a bunch is demonstrated. The damping rates are quantified for different modes and space-charge tune shifts. Finally, the head-tail instability with space charge is studied for the resistive-wall impedance below the mode coupling threshold. Results demonstrate that space-charge induced damping can suppress the instability for moderately strong space charge; instability growth rates saturate at strong space charge, in agreement with theoretical predictions.
Highlights
Space charge shifts the frequency of the incoherent betatron oscillations which can change stability thresholds, growth rates, and real eigenfrequencies of transverse coherent instabilities
For the nominal beam parameters of high-intensity synchrotrons, as the projected SIS100 of FAIR [1], or the CERN PS, the bunch conditions correspond to strong space charge, ÁQsc ) Qs, or moderate space charge, ÁQsc * Qs, where ÁQsc is the peak shift of the betatron tune due to space charge and Qs is the synchrotron tune
Numerical simulations appear to be indispensable for a comprehensive stability analysis in different beam parameter regimes and with various collective effects taken into account
Summary
Space charge shifts the frequency of the incoherent betatron oscillations which can change stability thresholds, growth rates, and real eigenfrequencies of transverse coherent instabilities. This is true for the head-tail instability of ion bunches in ring machines. In the case of a bunch, the synchrotron motion plays an important role and the space-charge tune spread due to the longitudinal density profile provides Landau damping. We demonstrate this Landau damping in particle tracking simulations and examine its role for the stability of head-tail modes at moderate and stronger space charge
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