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.

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

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

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

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

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

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

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

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

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

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. NuFact05, Frascati June 2005

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 NuFact05, Frascati June 2005

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

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