Thresholds of a bunched beam longitudinal instability in proton synchrotrons

The formulas and graphs for calculating instability thresholds arising during the interaction of a bunched proton beam with narrow-band resonator are given. The instabilities of three types with oscillations of a definite multipolarity, oscillations of some bound multipoles and with microwave oscillations arising as a result of addition of a great number of multipoles. The analysis of the above data shows that the increase of oscillations nonlinearity is accompanied by the growth of instability threshold only in the zone of separated and weakly bound multipoles. The increase of spread of synchrotron frequencies reduces the zone separated multipoles owing to which the microwave bunch instability can be caused by more and more low-frequency resonators. In the microwave zone practically there is no stabilizing effect of synchrotron frequencies spread. The instability threshold of the bunched beam now - where exceeds the microwave level