Linear dynamics of charged particles in the main lattices of storage rings

TL;DR

This paper explores the linear dynamics of charged particles in storage rings, deriving continuous solutions to Hill's equation and proposing a method to incorporate nonlinearities for better modeling of betatron oscillations.

Contribution

It introduces a novel approach to solving Hill's equation in storage rings using series expansion of magnetic fields, paving the way for more accurate particle motion analysis.

Findings

Derived continuous solutions to Hill's equation for storage rings

Developed a series expansion method for magnetic field components

Suggested incorporating nonlinearities for improved betatron oscillation modeling

Abstract

To study the characteristics of synchrotron radiation in magnetic fields of accelerators first the author was necessary to obtain a continuous solutions of Hill's equation. For this purpose the gradient or the components of magnetic field were developed in a series. The same procedure is followed now in the case of storage rings. This approach proved to be interesting not only from the point of view of describing the motion of partiles in ordinary three-dimensional space but also in the fact that we get new differential equations. This brief review can be regarded as an introduction to the proposed method. The next step may be to add nonlinearities. This would be the best approximation to the determination of betatron oscillations in the existing accelerators.