Synchrobetatron stop bands due to a single crab cavity

We analyze the stop band due to crab cavities for horizontal tunes that are either close to integers or close to half integers. The latter case is relevant for today’s electron/positron colliders. We compare this stop band to that created by dispersion in an accelerating cavity and show that a single typical crab cavity creates larger stop bands than a typical dispersion at an accelerating cavity. We furthermore analyze whether it is beneficial to place the crab cavity at a position where the dispersion and its slope vanish. We find that this choice is worth while if the horizontal tune is close to a half integer, but not if it is close to an integer. Furthermore we find that stop bands can be avoided when the horizontal tune is located at a favorable side of the integer or the half integer. While we are here concerned with the installation of a single crab cavity in a storage ring, we show that the stop bands can be weakened, although not eliminated, significantly when two crab cavities per ring are chosen suitably.