Lie Algebraic Methods for Treating Lattice Parameter Errors in Particle Accelerators

Abstract

Orbital dynamics in particle accelerators, and ray tracing in light optics, are examples of Hamiltonian systems. The transformation from initial to final phase space coordinates in such systems is a symplectic map. Lie algebraic techniques have been used with great success in the case of idealized systems to represent symplectic maps by Lie transformations. These techniques allow rapid computation in tracking particles while maintaining complete symplecticity, and easy extraction of analytical quantities such as chromaticities and aberrations. Real accelerators differ from ideal ones in a number of ways. Magnetic or electric devices, designed to guide and focus the beam, may be in the wrong place or have the wrong orientation, and they may not have the intended field strengths. The purpose of this dissertation is to extend the Lie algebraic techniques to treat these misplacement, misalignment and mispowering errors. Symplectic maps describing accelerators with errors typically have first-order terms. There are two major aspects to creating a Lie algebraic theory of accelerator errors: creation of appropriate maps and their subsequent manipulation and use. There are several aspects to the manipulation and use of symplectic maps. A first aspect is particle tracking. That is, one must find how particle positions are transformed by a map. A second is concatenation, the combining of several maps into a single map including nonlinear feed-down effects from high-order elements. A third aspect is the computation of the fixed point of a map, and the expansion of a map about its fixed point. For the case of a map representing a full turn in a circular accelerator, the fixed point corresponds to the closed orbit. The creation of a map for an element with errors requires the integration of a Hamiltonian with first-order terms to obtain the corresponding Lie transformation. It also involves a procedure for the complete specification of errors, and the generation of the map for an element with errors from the map of an ideal element. The methods described are expected to be applicable to other electromagnetic systems such as electron microscopes, and also to light optics systems.