Third Order Lie Map for the Sextupole Magnets

Abstract：

Sextupole magnets are usually used in beam analyzing systems with high resolving power, micro-beam systems and storage rings of high energy accelerators. It is well known that sextupoles are equivalent to the drift spaces under first order approximation. Therefore, this kind of optical elements is often used to correct the second order aberrations of beam optics systems. When it is necessary to calculate the third order aberrations of a system, or to correct the third order aberrations with octupole magnets, one should know the analytical expressions of the third order terms of sextupoles. In this paper, Lie algebraic methods were used in the analysis of relativistic particle trajectories in the sextupole magnets, and the solutions of third order approximation in the six dimensional phase spaces were obtained.