Energy limit of compact isochronous cyclotrons

TL;DR

This paper develops a relativistically accurate analytical model for transverse beam dynamics in compact isochronous cyclotrons, enabling precise determination of stability zones and energy limits at high energies.

Contribution

It introduces a practical, relativistically correct method with new formulas and graphs for analyzing beam stability and energy limits in cyclotrons, improving upon previous models.

Findings

Derived formulas for radial and vertical tunes considering relativistic effects.

Identified the stability zone and energy limit for various cyclotron configurations.

Validated the analytical model with simulations of the IBA C400 cyclotron.

Abstract

Existing analytical models for transverse beam dynamics in isochronous cyclotrons are often not valid or not precise for relativistic energies. The main difficulty in developing such models lies in the fact that cross-terms between derivatives of the average magnetic field and the azimuthally varying components cannot be neglected at higher energies. Taking such cross-terms rigorously into account results in an even larger number of terms that need to be included in the equations. In this paper, a method is developed which is relativistic correct and which provides results that are practical and easy to use. We derive new formulas, graphs and tables for the radial and vertical tunes in terms of the the flutter, its radial derivatives, the spiral angle and the relativistic gamma. Using this method, we study the $2\nu_r=N$ structural resonance ($N$ is number of sectors) and provide formulas and graphs for its stopband. Combining those equations with the new equation for the verical tune, we find the stability zone and the energy limit of compact isochronous cyclotrons for any value of N. We confront the new analytical method with closed orbit simulations of the IBA C400 cyclotron for hadron therapy.