Lawrence Berkeley National Laboratory

The space-charge driven envelope instability can be of great danger in high intensity accelerators and was studied using a two-dimensional (2D) envelope model and three-dimensional (3D) macroparticle simulations before. In this paper, we study the instability for a bunched beam using a three-dimensional envelope model in a periodic solenoid and radio-frequency (rf) focusing channel and a periodic quadrupole and rf focusing channel. This study shows that when the transverse zero current phase advance is below 90°, the beam envelope can still become unstable if the longitudinal zero current phase advance is beyond 90°. For the transverse zero current phase advance beyond 90°, the instability stopband width becomes larger with the increase of the longitudinal focusing strength and even shows different structure from the 2D case when the longitudinal zero current phase advance is beyond 90°. Breaking the symmetry of two longitudinal focusing rf cavities and the symmetry between the horizontal focusing and the vertical focusing in the transverse plane in the periodic quadrupole and rf channel makes the instability stopband broader. This suggests that a more symmetric accelerator lattice design might help reduce the range of the envelope instability in parameter space.