Abstract
The Lie map generator of the dipole fringe field is derived up to the 4th order of canonical variables. We discovered significant closed orbit deviation and octupolelike potential when the bending radius $\ensuremath{\rho}$ is small. We found that the closed orbit deviation is proportional to ${g}^{2}/\ensuremath{\rho}$ and the octupolelike potential effect is proportional to $1/(g{\ensuremath{\rho}}^{2})$, where $g$ is the vertical magnet gap.
Highlights
The fringe field of dipole magnets can be mportant in charged-particle beam dynamics [1,2]
Since a Lie map generator of an accelerator element is an effective Hamiltonian multiplied by the length of the element, we find the effective octupolelike potential of Eqs. (35) and (36) can be written as [18]
Two major findings were the closed orbit deviation, which is of the order of g2=ρ, and the octupolelike potential, which is of the order of 1=ðgρ2Þ
Summary
The fringe field of dipole magnets can be mportant in charged-particle beam dynamics [1,2]. An important feature of the dipole fringe field effect is the closed orbit deviation from the design orbit. This change of the closed orbit arises from the fact that the fringe field introduces continuously varying curvature, while the design orbit is defined by constant curvature starting from the hard edge dipole boundary This fact is naive and simple, its effect can be large for compact storage rings and should not be disregarded. The fringe field effects on nonlinear dynamics for compact rings and large emittance beams have been considered in Refs. This paper studies the effective thin map of the dipole soft fringe field using the Lie map method up to the leading order of Magnus’ series and up to the 4th order of canonical variables with respect to the design orbit.
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