Single bunch longitudinal instabilities in proton storage rings

In this paper we try to explain the single bunch energy spread increasing in a proton storage ring. Different from an electron machine, the random motions of some particles of a bunch are caused by the nonlinear synchrotron oscillations perturbed by the short range longitudinal wakefield. An analytical single bunch threshold current formula has been established to indicate the beginning of the longitudinal random motions. By using kinetic descriptions one shows that the energy spread of the particles which execute random motions will increase, and the energy spectrum width of the random motions will be enlarged with increasing bunch current.