1. Simulating transition crossing in the PS with HeadTail
E. Métral, G. Rumolo, B. Salvant
LIS meeting April 7th, 2008

2. Agenda Context of transition crossing in the PS
HeadTail Simulations at constant energies
HeadTail Simulations with acceleration
Longitudinal aspects
Crossing transition with a broadband impedance
Perspectives

3. Context of PS transition crossing
Elias presentation in January : Instabilities appear at transition crossing in the PS

4. Context of PS transition crossing
Question from PS2 project:
 how fast should we cross transition in PS2 to avoid instabilities ?
 Analytical calculations made by Elias (see 29/01/08 LIS meeting presentation )
 HeadTail was modified to enable acceleration during simulation

5. Simulation parameters
No broadband impedance
No space charge impedance
nTOF PS parameters (E. Métral)
Transition Energy is constant (t ~ 6.08)
Higher orders of momentum compaction factor are not taken into account
Beam momentum is increased linearly with time:
Here

6. Agenda Context of transition crossing in the PS
HeadTail Simulations at constant energies
HeadTail Simulations with acceleration
Longitudinal aspects
Crossing transition with a broadband impedance
Perspectives

7. BB – no SC – no acceleration – simulations at various energies
Bunch lengths and momentum spread were adjusted at each energy to get a linearly matched bunch in the PS bucket with a longitudinal emittance of 2 eVs.
Bunch Length
Emittance
Getting closer to transition
transition = 6.08
The bunch is longer far from transition  linear matching equations do not hold

8. BB – no SC – no acceleration – simulations at various energies
Analytical calculations of the longitudinal motion of a particle at the edge of a bunch with 2 eVs longitudinal emittance in a static bucket. For comparison, the accelerating bucket is also shown
Stationary bucket
2 eVs for a linear restoring force
2 eVs for non linear restoring force
Accelerating bucket
(the area of the (z,) ellipse is not constant, but these are not canonical variables)

9. BB – no SC – no acceleration – simulations at various energies
Instabilitity threshold is at  ~ 5.2

10. With BB no acceleration -  = 5.2 -  = -0.010

11. With BB no acceleration -  = 5.4 -  = -0.007

12. With BB no acceleration -  = 5.6 -  = -0.005

13. With BB no acceleration -  = 5.8 -  = -0.003

14. With BB no acceleration -  = 6.0 -  = -0.0008

15. Comparison with analytical estimates

16. Agenda Context of transition crossing in the PS
HeadTail Simulations at constant energies
HeadTail Simulations with acceleration
Longitudinal aspects
Crossing transition with a broadband impedance
Perspectives

17. Longitudinal ellipse shape while crossing transition
Elias’s plot of the tilt of the ellipse in the longitudinal phase space
Case of the nTOF bunch in the PS WITHOUT space charge (2/2)

18. Checking simulated longitudinal motion during transition crossing
HeadTail simulation : observing 10% of the macroparticles in the longitudinal phase space
What about the shape of the ellipse?  follow some particles over 10 turns…

19. Initial distribution
Only the blue particles will be displayed to assess the tilt of the ellipse

20. Tracking macroparticles with l ~ 2 eVs

21. No BB – no SC – comparing HeadTail with analytical calculations from Elias
Reasonable agreement

22. Agenda Context of transition crossing in the PS
HeadTail Simulations at constant energies
HeadTail Simulations with acceleration
Longitudinal aspects
Crossing transition with a broadband impedance
Perspectives

23. BB impedance – no SC
Initial  = 5.65
Unstable from t=0

24. BB impedance – no SC  Linear scale <y> Log scale <y>
Initial  = 4.6
Stable at t = 0
Unstable at
 = 5.25 
Relativistic gamma
Linear scale <y>
Vertical position of Centroid [m]
Log scale <y>
Vertical position of Centroid [m]
Growth rate
~ 60 µs

25. Agenda Context of transition crossing in the PS
HeadTail Simulations at constant energies
HeadTail Simulations with acceleration
Longitudinal aspects
Crossing transition with a broadband impedance
Perspectives

26. Perspectives
Good agreement between analytical calculations and HeadTail simulations
In HeadTail simulations
Include possibility of t jump
Include transverse space charge
Take into account the non linear momentum factor 1
Include possibility of longitudinal space charge
Simulate PS2 parameters and assess the maximum time allowed for transition crossing