Analytical Solution of Transverse Oscillation in Cyclotron Using LP Method
Kai Zhou, Yuntao Song, Kaizhong Ding, Jian Ge, Kai Yao
TL;DR
This paper presents an approximate analytical solution for transverse oscillations in cyclotrons influenced by magnetic fields, using the Lindstedt-Poincare method, and verifies its accuracy against numerical methods.
Contribution
It introduces a first-order analytical solution for transverse oscillations in cyclotrons with spiral sectors, enhancing understanding of magnetic field effects.
Findings
Analytical solutions closely match numerical results.
Effective for isochronous cyclotrons with spiral sectors.
Provides insights into oscillations at specific velocities.
Abstract
We have carried out an approximate analytical solution to precisely consider the influence of magnetic field on the transverse oscillation of particles in cyclotron. The differential equations of transverse oscillation are solved from the Lindstedt-Poincare method. After careful deduction, the accurate first order analytic solutions are obtained. The analytical solutions are applied to the magnetic field, comes from an isochronous cyclotron with four spiral sectors, the accuracy of these analytical solutions is verified and confirmed from the comparison of numerical method. Finally, we discussed the transverse oscillation at v0=N/2 , using the same analytical solution.
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