Three-dimensional envelope instability model in periodic focusing channels

TL;DR

This paper develops a 3D envelope instability model for high-intensity accelerators, revealing how longitudinal and transverse focusing parameters influence instability regions and suggesting symmetric lattice designs for mitigation.

Contribution

It introduces a comprehensive 3D model for envelope instability in periodic focusing channels, extending previous 2D and simulation studies with new insights on parameter effects.

Findings

Instability occurs when longitudinal phase advance exceeds 90 degrees, even if transverse phase advance is below 90 degrees.

Increasing longitudinal focusing strength broadens the instability stopband when transverse phase advance is above 90 degrees.

Symmetry breaking in RF cavities and focusing elements enlarges the instability region.

Abstract

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 propose a three-dimensional envelope instability model to study the instability for a bunched beam 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 $90$ degrees, the beam envelope can still become unstable if the longitudinal zero current phase advance is beyond $90$ degrees. For the transverse zero current phase advance beyond $90$ degrees, 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 $90$ degrees. 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.