1. PyECLOUD G. Iadarola, G. Rumolo Thanks to: F. Zimmermann, G. Arduini, H. Bartosik, C. Bhat, O. Dominguez, M. Driss Mensi, E. Metral, M. Taborelli

2. The electron cloud buildup Proton bunch e - is emitted (photoelectric effect)

3. The electron cloud buildup Proton bunch e - is emitted (photoelectric effect) Secondary Electron Emission SEY :

4. The electron cloud buildup If the SEY is large enough, Secondary Electron Emission can drive an avalanche multiplication effect filling the beam chamber with an electron cloud Proton bunch e - is emitted (photoelectric effect)

5. The electron cloud buildup The electron density grows exponentially during the passage of the bunch train and then saturates (due to space charge effects within the cloud) The trailing bunches of the train see the largest electron density

6. Ingredients for e-cloud build-up simulation 1. Seed electrons generation (gas ionization, photoemission) 2. Force exerted by the beam on e - 3. Force exerted by the e - on each other (space charge) 4. Equations of motion (also in presence of an external magnetic field) 5. Secondary emission

7. t=t+Δt Evaluate the electric field of beam at each MP location Generate seed e - Compute MP motion (t->t+Δt) Detect impacts and generate secondaries PyECLOUD flowchart Evaluate the e - space charge electric field PyECLOUD is a 2D macroparticle (MP) code for the simulation of the electron cloud build-up with: Arbitrary shaped chamber Ultra-relativistic beam Externally applied (arbitrary) magnetic field

8. t=t+Δt Evaluate the electric field of beam at each MP location Generate seed e - Compute MP motion (t->t+Δt) Detect impacts and generate secondaries Evaluate the e - space charge electric field Evaluate the number of seed e - generated during the current time step and generate the corresponding MP: Residual gas ionization and photoemission are implemented PyECLOUD flowchart

9. t=t+Δt Evaluate the electric field of beam at each MP location Generate seed e - Compute MP motion (t->t+Δt) Detect impacts and generate secondaries Evaluate the e - space charge electric field The field map for the relevant chamber geometry and beam shape is pre-computed on a suitable rectangular grid and loaded from file in the initialization stage When the field at a certain location is needed a linear (4 points) interpolation algorithm is employed PyECLOUD flowchart

10. t=t+Δt Evaluate the electric field of beam at each MP location Generate seed e - Compute MP motion (t->t+Δt) Detect impacts and generate secondaries Evaluate the e - space charge electric field Classical Particle In Cell (PIC) algorithm: Electron charge density distribution ρ(x,y) computed on a rectangular grid Poisson equation solved using finite difference method Field at MP location evaluated through linear (4 points) interpolation PyECLOUD flowchart

11. t=t+Δt Evaluate the electric field of beam at each MP location Generate seed e - Compute MP motion (t->t+Δt) Detect impacts and generate secondaries Evaluate the e - space charge electric field When possible, “strong B condition” is exploited in order to speed-up the computation The dynamics equation is integrated in order to update MP position and momentum: PyECLOUD flowchart

12. t=t+Δt Evaluate the electric field of beam at each MP location Generate seed e - Compute MP motion (t->t+Δt) Detect impacts and generate secondaries Evaluate the e - space charge electric field When a MP hits the wall theoretical/empirical models are employed to generate charge, energy and angle of the emitted charge According to the number of emitted electrons, MPs can be simply rescaled or new MP can be generated PyECLOUD flowchart

13. Macroparticle size management In an electron-cloud buildup, due to the multipacting process, the electron number extends over several orders of magnitude It is practically impossible to choose a MP size that is suitable for the entire simulation (allowing a satisfactory description of the phenomenon and a computationally affordable number of MPs) 72 bunches – 25ns spac.

14. Macroparticle size management x A reference MP size N ref is used to “take decisions”: 1)Seed MP generation: the generated MPs have size N ref 2)Secondary MP emission: additional true secondary MPs are emitted if the total emitted charge is >1.5N ref 3)MP cleaning: at each bunch passage a clean function is called to eliminate all the MPs with charge <10 -4 N ref 72 bunches – 25ns spac.

15. Macroparticle size management 10. MP ref. size [m -1 ] The reference MP size N ref is adaptively changed during the simulation:

16. Macroparticle size management 10. MP ref. size [m -1 ] The reference MP size N ref is adaptively changed during the simulation:

17. Macroparticle size management 10. MP ref. size [m -1 ] The reference MP size N ref is adaptively changed during the simulation:

18. Macroparticle size management 10. MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

19. Macroparticle size management 10. MP ref. size [m -1 ] MP reg. MP set regeneration a.Each macroparticle is assigned to a cell of a uniform grid in the 5-D space (x,y,v x,v y,v z ) obtaining an approximation of the phase space distribution b.The new target MP size is chosen such that: c.A new MPs set, having the new reference size, is generated according to the computed distribution The error on total charge and total energy does not go beyond 1-2%

20. Macroparticle size management 10.45. MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

21. Macroparticle size management 10.45. MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

22. Macroparticle size management 10.45. MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

23. Macroparticle size management 10.45.2.1e2 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

24. Macroparticle size management 10.45.2.1e2 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

25. Macroparticle size management 10.45.2.1e21.1e3 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

26. Macroparticle size management 10.45.2.1e21.1e3 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

27. Macroparticle size management 10.45.2.1e21.1e35.5e3 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

28. Macroparticle size management 10.45.2.1e21.1e35.5e3 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

29. Macroparticle size management 10.45.2.1e21.1e35.5e32.9e4 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

30. Macroparticle size management 10.45.2.1e21.1e35.5e32.9e4 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

31. Macroparticle size management 10.45.2.1e21.1e35.5e32.9e41.3e5 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

32. Macroparticle size management 10.45.2.1e21.1e35.5e32.9e41.3e5 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

33. Macroparticle size management 10.45.2.1e21.1e35.5e32.9e41.3e53.1e5 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

34. Macroparticle size management 10.45.2.1e21.1e35.5e32.9e41.3e53.1e5 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

35. Macroparticle size management 10.45.2.1e21.1e35.5e32.9e41.3e53.1e5 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

36. Macroparticle size management 10.45.2.1e21.1e35.5e32.9e41.3e53.1e5 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

37. Macroparticle size management 10.45.2.1e21.1e35.5e32.9e41.3e53.1e5 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

38. Macroparticle size management 10.45.2.1e21.1e35.5e32.9e41.3e53.1e53.6e5 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

39. Macroparticle size management 10.45.2.1e21.1e35.5e32.9e41.3e53.1e53.6e5 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

40. Macroparticle size management 10.45.2.1e21.1e35.5e32.9e41.3e53.1e53.6e5 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation:

41. Macroparticle size management 10.45.2.1e21.1e35.5e32.9e41.3e53.1e53.6e5 MP ref. size [m -1 ] MP reg. The reference MP size N ref is adaptively changed during the simulation: