Linear Beam Stability in Periodic Focusing Systems:Krein Signature and Band Structure

TL;DR

This paper investigates linear beam stability in periodic focusing systems using Krein signature and band structure concepts, providing new insights into eigenvalue behavior and envelope instability analysis.

Contribution

It introduces the application of Krein theory to envelope instability analysis and explores the role of eigenmode collisions in beam instability.

Findings

Eigenvalues and characteristics of one-period maps were numerically calculated.

Instabilities arise from eigenmode collisions of opposite Krein signatures.

Band gaps are associated with the formation of instabilities.

Abstract

The general question of how a beam becomes unstable has been one of the fundamental research topics among beam and accelerator physicists for several decades. In this study, we revisited the general problem of linear beam stability in periodic focusing systems by applying the concepts of Krein signature and band structure. We numerically calculated the eigenvalues and other associated characteristics of one-period maps, and discussed the stability properties of single-particle motions with skew quadrupoles and envelope perturbations in high-intensity beams on an equal footing. In particular, an application of the Krein theory to envelope instability analysis was newly attempted in this study. The appearance of instabilities is interpreted as the result of the collision between eigenmodes of opposite Krein signatures and the formation of a band gap.