Beam Optics and Dynamics of FFAG Accelerators

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Beam Optics and Dynamics of FFAG Accelerators

• Shinji Machida and Etienne Forest
• KEK
• 2001/5/25 _at_Tsukuba

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Orbit in FFAG (radial sector)
Fixed Field Alternating Gradient
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Optics design means

• Orbit calculation
• Based on 3D field mapping data.
• Estimate effects of nonlinearity and dynamic
  aperture
• Systematic study of dependence on
• momentum,
• betatron amplitude,
• magnet strength.
• Dynamic aperture as all included.

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Modeling procedure

• Take only a linear part of a gradient magnet
  along reference orbit.
• Synchrotron design code such as SAD can be
  utilized.
• Make a 3D field mapping data with TOSCA and track
  a particle (step by step integration.)
• Brute force integration such as Runge-Kutta
  method.
• (Or) Make a 3D field mapping data and construct a
  map using TPSA (trancated poly. series algeb.)
• Use a code such as PTC by Forest written in F90.

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Linearized modeling

• Assumption
• Constant field in F and D.
• Closed orbit is described as an arc.
• Along the orbit, linear gradient term is
  constant.
• Edge focusing and fringe field are taken into
  account.
• No nonlinearity.

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0.3 to 1 GeV/c m acceleration ring

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Modeling procedure

• Take only a linear part of a gradient magnet
  along reference orbit.
• Synchrotron design code such as SAD can be
  utilized.
• Make a 3D field mapping data with TOSCA and track
  a particle (step by step integration.)
• Brute force integration such as Runge-Kutta
  method.
• Make a 3D field mapping data and construct a map
  using TPSA (trancated poly. series algeb.)
• Use a code such as PTC by Forest written in F90.

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Replace FFAG with a transformation map

• Each trapezoid shows one sector.
• Track a polymorphic variable (real and Taylor
  series) through one sector.
• 3D field mapping is converted to a transformation
  map Normal Form.

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Each sector consist of many slices in z (or q)
direction.
E. Forest at FFAG2000
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Tracking in a magnet

• 3D field data (with TOSCA) is fitted by
  orthogonal functions (Legendre polynomial)
  globally or by locally in each slice.
• Step by step integration of polymorphic variable
  with 1st or 4th order Runge-Kutta method.
• If necessary, symplectic condition is enforced.

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Calculate beta functions

• The Normal Form is given by
• where M is the one-turn (one-section) map, A is
  the normalizing transformation, and R is a
  rotation. The beta function can be computed as
• when there is no coupling, is zero.

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Estimate parameter dependence

• Higher order terms of the Normal Form give
  parameter dependence of
• phase space coordinates
• other machine parameters such as the strength of
  field.

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(old) Step by step integration as a comparison

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Summary

• Nonlinearity is inherent in a FFAG accelerator.
  Even a closed orbit is not determined without it.
• Although a linearized model is useful for a
  designer to start with, optics design with
  nonlinearity is a must before an actual
  construction.
• The PTC (Polymorphic Tracking Code) is just an
  ideal tool to study FFAG (and other machines even
  cyclotron.)