Stability Criteria for High-Intensity Single-Bunch Beams in Synchrotrons

In circular accelerators, the interaction between the beam and its surroundings is described in terms of bunched-beam modes through Sacherer's formalism. As the bunch intensity increases, the different modes, separated by the synchrotron frequency, can no longer be treated separately. In this regime, the wake fields couple the modes together and a wave pattern travelling along the bunch is created. It leads to the longitudinal and transverse mode-coupling instabilities, aspects of which are discussed under many different names in a variety of different papers. The present work reviews and unifies these approaches. It is shown that for the transverse plane, the same intensity threshold is obtained through 5 seemingly diverse formalisms: (i) transverse mode coupling, (ii) beam break-up, (iii) fast blow-up, (iv) post head-tail, and (v) a quasi coasting-beam approach using the peak values of bunch current and momentum spread as input for the coasting-beam formula. For the longitudinal plane, a new stability criterion is derived using the mode-coupling formalism. This formula is close to the Keil-Schnell-Boussard criterion, when the space-charge impedance is much smaller than the resonator one.