Choice of harmonic number for the ISABELLE accelerating rf system

Originally, h = 2 was chosen for the accelerating rf system to avoid growth of coupled bunch mode longitudinal instabilities. The ability to operate ISABELLE in a bunched mode and maintaining six interaction points plus the compatibility with the boxcar transfer scheme suggested for transfer from an eventual future accumulator ring has made it desirable to choose a different harmonic number, namely h = 3. It is shown in the following that the impedance threshold for these instabilities is higher than the impedance limit required during stacking to obtain design performance. The threshold is independent of the harmonic number, so the choice of h is free from an instability point of view, as long as we are below this threshold. For h = 3, the required peak voltage to produce the same acceptance and acceleration rate as the h = 2 system is 35 kV compared to 30 kV for the h = 2 system. The total stored energy in the h = 3 cavities will be less than in h = 2 cavities, so the cost is roughly unchanged. For the above mentioned reasons it is strongly recommended to choose h = 3