Simulations of sawtooth instability

The equilibrium self-consistent distribution of particles in a high intensity electron synchrotron can be found using the Haissinski equation and the wake field. At some threshold intensity the bunch becomes unstable. However, radiation damping causes the particles to be confined and the instability does not necessarily cause loss of particles. It was observed in simulations with a very simple wake field and short bunches, that energy spread and bunch length oscillate in a sawtooth fashion. We find that this is due to the double-peaked nature of the stationary distribution. Over many synchrotron oscillations, particles diffuse from the head peak to the tail to the point where the tail peak becomes as large as the head. The two resulting sub-bunches then collapse together in less than one synchrotron oscillation, causing a net blow-up in emittance. Radiation damping reduces the emittance and diffusion begins again