On the derivation of Amplitude Detuning and Chromaticity Formulas for Particle Accelerators

TL;DR

This paper provides a detailed derivation of amplitude detuning and chromaticity formulas in particle accelerators, including effects of multipoles, feed-down, and AC-Dipoles, starting from fundamental principles.

Contribution

It offers a comprehensive, step-by-step derivation of amplitude detuning formulas up to dodecapolar order, including new insights on feed-down and AC-Dipole effects.

Findings

Derived formulas for amplitude detuning from multipoles up to dodecapolar order.

Explored the principle of feed-down with example terms.

Included the influence of AC-Dipole in amplitude detuning.

Abstract

This note aims to provide a complete and comprehensible derivation of the contributions to amplitude detuning in first order from multipoles up to dodecapolar order. In addition, the principle of feed-down is explored and example terms are given. The influence of an AC-Dipole is also included in the amplitude detuning. The derivation starts from very basic principles and maths useful for the derivation, such as Taylor and Binomial expansion is given at the beginning of the note.