Revised Hamiltonian near Third-integer Resonance and its dynamics in Electron Storage Ring Division of Advanced Nuclear Engineering Pohang University of Science and Technology

Abstract

The third-integer resonance is a critical phenomenon in accelerator physics, significantly impacting the stability and performance of electron storage rings. Understanding and controlling the dynamics near this res- onance are essential for optimizing beam quality and minimizing particle loss in high-energy accelerators. This study aims to develop a comprehen- sive theoretical and practical framework for analyzing and mitigating the effects of the third-integer resonance on particle dynamics. Several mathematical techniques, including the Lie perturbation theory and elliptic integral, are employed to investigate the dynamics near the third-integer resonance. By deriving the time-independent Hamiltonian and applying perturbative methods, we systematically analyze the influence of resonance on electron phase space. Numerical simulations complement the theoretical analysis, validating the models and providing practical insights into the behavior of particle beams under resonant conditions. The study successfully derives the Hamiltonian for an electron storage ring near the third-integer resonance, capturing the essential dynamics of the system. An extended phase space analysis transforms the time-dependent Hamiltonian into a time-independent problem, simplifying the investigation of resonant effects. Lie perturbation theory reveals the impact of small per- turbations on the system, providing corrections to the dynamics. Numerical tracking results validate the theoretical models, demonstrating the accuracy of the derived Hamiltonian and the perturbative methods. The study also uncovers the devil’s staircase phenomenon, offering insights into its impli- cations for beam stability. The results highlight the significance of understanding and managing the third-integer resonance in electron storage rings. The derived theoretical models and numerical simulations provide a robust framework for predicting and controlling resonant effects, contributing to the design of more stable and efficient accelerators.