Progress in Code Benchmarking Shinji Machida CCLRC/RAL/ASTeC 30 March, 2006 http://hadron.kek.jp/~machida/doc/spacecharge/ gsi_mar2006/machida_20060330.ppt & pdf

Amplitude growth by trapping in moving islands Proposed by G. Franchetti and I. Hofmann in 2002. Very nice model and a strong candidate of halo formation mechanism. Simpsons with frozen space charge model shows similar behavior.

“Adiabatic” and “Scattering” regime When synchrotron tune is very low (1/15000), a particle follows the islands oscillations. “adiabatic” regime When synchrotron tune is medium (1/1000), particle behavior becomes more stochastic. “scattering” regime Benchmarking becomes difficult in the latter regime.

Benchmarking with future experiments What we would hopefully measure is population of halo particles or change of tail distribution. Unfortunately, not single particle trajectory. Data in experiment is multi-particle quantities.

Existing models of particle trapping A.W.Chao and M.Month, NIM 121, 1974 Beam-beam effects creates nonlinear resonance, 5th order. Magnet ripple and synchrotron oscillations introduce tune modulation. R. Cappi and M. Giovannozzi, PRL 88, 2002 For extraction Controlled nonlinearity and tune sweep. M. Aiba, et. al., to be PRSTAB, 2006 Single crossing in FFAG due to not-perfect scaling magnet. FFAG has all order of nonlinearity. Theory tells “trapping efficiency” that is a statistical measure.

Multi-particle tracking and its statistical analysis are the next step of benchmarking. code vs. code code vs. theory code vs. experiment

Relevant theory Aiba extends the Chao and Month model to 3rd order resonance and confirms it experimentally using PoP FFAG. Trapping efficiency for 3rd order resonance is total area of islands : “adiabatic parameter” or normalized crossing speed as：beam emittance of island center : crossing speed DNL : nonlinear tune shift De : resonance width (from Aiba’s Ph.D. thesis)

Comparison with experiments Trapping efficiencies Efficiency in experiment (proportional to crossing speed) * k are about 3. (from Aiba’s Ph.D. thesis)

Criterion to avoid trapping Adiabatic parameter or normalized crossing speed should be more than 7. (from Aiba’s Ph.D. thesis)

Another way of looking at trapping in simulation Take a initial distribution (not a particle) in the following way. See the evolution of horizontal rms emittance. Detailed distribution can be tracked. px dp/p x f

Parameters in simulation Nonlinear error Synchrotron tune : Space charge tune spread

ns dependence, for example k2=0.1, ns=0.0002 k2=0.1, ns=0.0005 k2=0.1, ns=0.001 k2=0.1, ns=0.002

Remark when we compare theory and simulation Single crossing theory explains the (only) first increase of rms emittance. Both theory and simulation do not have collective effects.

Code vs. theory (1) Synchrotron tune is proportional to crossing speed.

With same adiabatic parameter k2=0.02, ns=0.0002 k2=0.05, ns=0.0005 k2=0.1, ns=0.001 k2=0.2, ns=0.002

Code vs. theory (2) Keep a1 (adiabatic parameter or normalized crossing speed) constant and see A (island area) dependence.

Next step Quantitative estimate of parameters has not been done yet. If we can establish the connection among theory, simulation and experiment, following argument would be possible. “In order to keep halo level less than 10E-X, adiabatic parameter should be more than Y. Therefore, when the synchrotron tune is ns , initial emittance is e , and space charge tune spread is Dn, tolerable resonance width is less than Z.”