Lie Algebraic Methods for Treating Lattice Parameter Errors in Particle Accelerators

Lie Algebraic Methods for Treating Lattice Parameter Errors in Particle Accelerators

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1986

##### Authors

Healy, Liam Michael

##### Advisor

Dragt, Alex J.

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##### 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.