1. Progress in Code Benchmarking
Shinji Machida
CCLRC/RAL/ASTeC
30 March, 2006
gsi_mar2006/machida_ ppt & pdf

2. 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.

3. “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.

4. 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.

5. 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.

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

7. 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)

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

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

10. 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

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

12. 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

13. 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.

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

15. 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

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

17. 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.”