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

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

Design of the isochronous cell [Ref.G.Rees] NuFact05, Frascati 23/06/2005 Original design L cell = 0.65 m BF is a multipole gradient is dB/dx. 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 2: The electron model isochronous cell. Figure 3: The sector magnet in the zgoubi model

Field models NuFact05, Frascati 23/06/2005 Gradient profiles K(m -2 ) vs. x(m) The magnets’ gradient are constitutive of the design data, they are approximated using 4th degree polynomials. The gradients are integrated to get the multipole coefficients of the field, as needed in the zgoubi data file. bd BD BF

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

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

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 NuFact05, Frascati June 2005 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 = x co )

Amplitude detuning NuFact05, Frascati 23/06/2005 Horizontal Vertical 20 MeV 17 MeV 15.8 MeV 14 MeV 12.2 MeV 11 Mev

Beam transmission (1) NuFact05, Frascati June 2005 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.

Beam transmission (2) NuFact05, Frascati June 2005 A particles beam is launched for 15 turn acceleration inside the acceptances obtained with the previous run.  x  11.3  mm mrad   x  normalised = 243  mm mrad  z = 10.4  mm mrad  z, normalised = 224  mm mrad

Tunes : acceleration cycle NuFact05, Frascati June 2005 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

Summary NuFact05, Frascati 23/06/2005 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