Single-particle dynamics in theoretical minimum emittance cell

Abstract

We have developed a parametrization of the linear optics in a theoretical minimum emittance cell with three quadrupoles. It consists of five independent parameters: two phase advances, a dimensionless horizontal beta function at the center of the dipole, and a bending angle ϕ and a length L of the dipole. For the zero chromaticity cell, we again find that the dynamic aperture in the normalized phase space is scaled according to, A[over ¯]∝ϕsqrt[L]. Moreover, we study nonlinear dynamics near the third-order coupled resonances in the framework of the resonance normal form. In particular, we derive the effective Hamiltonians using the Lie algebra method and show that the periodic orbits in tracking can be interpreted as solutions of the Hamilton’s equations. Surprisingly, we discover that the scheme is not only applicable to the single resonance but also to the double resonances.