Abstract
Fringe fields in multipole magnets can have a variety of effects on the linear and nonlinear dynamics of particles moving along an accelerator beam line. An accurate model of an accelerator must include realistic models of the magnet fringe fields. Fringe fields for dipoles are well understood and can be modeled at an early stage of accelerator design in such codes as mad8, madx, gpt or elegant. Existing techniques for quadrupole and higher order multipoles rely either on the use of a numerical field map, or on a description of the field in the form of a series expansion about a chosen axis. Usually, it is not until the later stages of a design project that such descriptions (based on magnet modeling or measurement) become available. Furthermore, series expansions rely on the assumption that the beam travels more or less on axis throughout the beam line; but in some types of machines (for example, Fixed Field Alternating Gradients or FFAGs) this is not a good assumption. Furthermore, some tracking codes, such as gpt, use methods for including space charge effects that require fields to vary smoothly and continuously along a beam line: in such cases, realistic fringe field models are of significant importance. In this paper, a method for constructing analytical expressions for multipole fringe fields is presented. Such expressions allow fringe field effects to be included in beam dynamics simulations from the start of an accelerator design project, even before detailed magnet design work has been undertaken. The magnetostatic Maxwell equations are solved analytically and a solution that fits all orders of multipoles is derived. Quadrupole fringe fields are considered in detail as these are the ones that give the strongest effects. The analytic expressions for quadrupole fringe fields are compared with data obtained from numerical modeling codes in two cases: a magnet in the high luminosity upgrade of the Large Hadron Collider inner triplet, and a magnet in the nonscaling FFAG EMMA. In both examples, the analytical expressions provide a good approximation to the numerical field maps.
Highlights
Fringe fields in multipole magnets can have a variety of effects on the linear and nonlinear dynamics of particles moving along an accelerator beam line
Series expansions rely on the assumption that the beam travels more or less on axis throughout the beam line; but in some types of machines this is not a good assumption
A method for constructing analytical expressions for multipole fringe fields is presented. Such expressions allow fringe field effects to be included in beam dynamics simulations from the start of an accelerator design project, even before detailed magnet design work has been undertaken
Summary
Fringe fields represent regions that lie at the edges of a magnet where there is a transition from the nominal field to zero field, or to the field in an adjacent magnet. For quadrupoles and higher-order multipoles, the fields in the fringe region have three components, and analytical expressions. The implementation of fringe fields is important in some tracking codes which include effects such as space charge, as is the case in GPT (General Particle Tracer) [1], for example. The results obtained in this paper make possible an alternative method, based on analytical expressions for the fields as functions of position, which provide exact solutions to the static Maxwell equations in three dimensions. This allows for arbitrarily smooth fields to be constructed and used in tracking codes as well as the possible creation of transfer maps.
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