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

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.