Longitudinal instability in high-intensity synchrotron including the perturbation of synchronous phase

In a high-intensity synchrotron, both amplitude and frequency of radio frequency (RF) acceleration voltage are modulated. In that case, the particle on a synchronous phase of the unperturbed RF voltage is not given the equilibrium energy that is related to a magnetic guide field. Including this difference in the given energy from the equilibrium one, we suggest that the equilibrium particle can revolve stably on the closed orbit, which is different from a nominal orbit without a modulation of RF voltage. And then, we formulate a new equation of motion and examine the Robinson instability. Here, the effect of beam intensity modulation is also included. This proposed method results in the fact that the stability criterion is significantly different from the conventional one, and requires optimum parameters of an RF system to ensure stability.