6-D dynamics in an isochronous FFAG lattice e-model

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6-D dynamics in an isochronous FFAG
latticee-model
Franck lemuet, Doctoral Student
CERN/CEA NuFact05, Frascati 24/06/2005

• Main topic
• Tracking code development
• 3-D simulation of the field in an isochronous
  FFAG optics.
• Application to study of beam transmission by
  multiturn tracking.

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The ray-tracing method Ingredients for magnet
simulation
Integration of the Lorentz equation, based on
Taylor series expansions
Figure 1 Position and velocity of a particle in
the reference frame.
NuFact05, Frascati 23/06/2005
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Design of the isochronous cell Ref.G.Rees

• Original design
• Lcell 0.65 m
• BF is a multipole
• gradient is dB/dx.

Figure 2 The electron model isochronous cell.

• Zgoubi model
• BF magnet is a sector magnet
• gradient is dB/dr
• sector angle value of half the cell deviation
• ? ?1/2 (360/45) 4 deg

Figure 3 The sector magnet in the zgoubi model
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Field models
Gradient profiles K(m-2) vs. x(m)
bd
The magnets gradient are constitutive of the
design data, they are approximated using 4th
degree polynomials.
BF
The gradients are integrated to get the multipole
coefficients of the field, as needed in the
zgoubi data file.
BD
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Tracking in a cell

• T.O.F in a cell is 2.177 ns

Isochronicity is better than a ps

• Tunes in a cell

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Tracking in a cell vertical field
Fringe field model
No overlapping
Sharp edge model
Vertical kick correction
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Tracking in a cell closed orbits
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New fitting procedures

• We have enhanced fitting capabilities in Zgoubi
  in relation to FFAG design, for instance allow
  automatic adjustement of bis coefficients
• so as to match tunes, or isochronism, etc
• Automatic search of closed orbits and Twiss
  parameters, for given set of energies
• Etc

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Stability limits
Two goals 1. Check symplecticity of the motion
over the all energy span. 2. Find the maximum
stable amplitudes in both planes, as well as
coupled
Pure and coupled horizontal motion limits
Vertical motion limits (x xco)
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Amplitude detuning
Horizontal
Vertical

• 20 MeV
• 17 MeV
• 15.8 MeV
• MeV
• 12.2 MeV
• 11 Mev

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Beam transmission (1)
A 10000 particles beam is launched for 15 turn
acceleration (45 cells/turn), from 11 to 20
MeV. 40 kV per cavity, no synchrotron motion.
Cavities are put every three cells at the center
of the long drift.
Initial phase space
Envelopes
Initial phase spaces of transmited particles
after the acceleration cycle.
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Beam transmission (2)
A 10000 particles beam is launched for 15 turn
acceleration inside the acceptances obtained
with the previous run.
ex 11.3 p mm mrad ex,normalised
243 p mm mrad ez 10.4 p mm mrad
ez,normalised 224 p mm mrad
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Tunes acceleration cycle

• A particle is launched at the injection energy
  (11 MeV) on its closed orbits.
• Tunes are computed during the acceleration
  cycle, approximate to paraxial tunes due to the
• low energy detuning .

Extraction 20 MeV
Injection 11 MeV
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Summary

• Efficient tools for tracking studies has been
  developed
• To do
• Focus on beam losses and correlations with
  resonnances crossing
• Study the new design of the e-model with
  insertions

NuFact05, Frascati 23/06/2005