1. 6-D dynamics in an isochronous FFAG lattice e-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.

2. 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

3. 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
θ = 1/2 (360/45) = 4 deg
Figure 3: The sector magnet in the zgoubi model
NuFact05, Frascati 23/06/2005

4. Field models bd BF BD Gradient profiles K(m-2) vs. x(m)
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
NuFact05, Frascati 23/06/2005

5. Tracking in a cell T.O.F in a cell is 2.177 ns
Isochronicity is better than a ps
Tunes in a cell
NuFact05, Frascati June 2005

6. Tracking in a cell : vertical field
Fringe field model
No overlapping
Sharp edge model
Vertical kick correction
NuFact05, Frascati 23/06/2005

7. Tracking in a cell : closed orbits
NuFact05, Frascati 23/06/2005

8. New fitting procedures
We have enhanced fitting capabilities in Zgoubi in relation to FFAG design, for instance allow automatic adjustement of bi’s coefficients
so as to match tunes, or isochronism, etc …
Automatic search of closed orbits and Twiss parameters, for given set of energies
Etc …
NuFact05, Frascati 23/06/2005

9. 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)
NuFact05, Frascati June 2005

10. Amplitude detuning 20 MeV 17 MeV 15.8 MeV MeV 12.2 MeV 11 Mev
Horizontal
Vertical
20 MeV
17 MeV
15.8 MeV
MeV
12.2 MeV
11 Mev
NuFact05, Frascati 23/06/2005

11. Beam transmission (1)
A 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.
NuFact05, Frascati June 2005

12. Beam transmission (2)
A particles beam is launched for 15 turn acceleration inside the acceptances obtained with
the previous run.
ex = p mm mrad ex,normalised = 243 p mm mrad
ez = p mm mrad ez,normalised = 224 p mm mrad
NuFact05, Frascati June 2005

13. 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 .
Injection
11 MeV
Extraction
20 MeV
NuFact05, Frascati June 2005

14. Efficient tools for tracking studies has been developed
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