Simulation of coupled bunch mode growth driven by a high-Q resonator: a transient response approach

The use of a longitudinal phase-space tracking code, ESME, to simulate the growth of a coupled-bunch instability in the Fermilab Booster is examined. The calculation of the resonant response is described. Two simulations are presented. In the first, all 84 RF buckets are populated with a bi-Gaussian distribution of emittance of 0.02 eV-s. Each bunch is represented by 100 macro-particles. The entire distribution (the ring) is divided into 3000 bins for the purposes of computing and applying the voltage due to the resonance. The charge corresponding to each bunch is set to 6.6×10¹² protons. In the second simulation, the parameters are the same as in the first, except for the fact that five buckets are not initially populated. The absence of these bunches enhanced the growth of the mode considerably. The induced voltage in the second case rises to an amplitude an order of magnitude larger than the first, 200 kV vs. 7 kV. The larger growth of the mode in the gapped-beam case may be caused by the enhancement of the harmonic component of the current at mode 56 initially, whereas in the symmetrical case the instability grows out of noise