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BBU simulation

BBU simulation. William Lou Supervisor: Prof. Georg Hoffstaetter. BBU overview BBU theory Bmad simulation 2) CBETA BBU results What is CBETA HOM assignment statistics (1-pass and 4-pass) Aim for higher Results with additional phase-advances Results with x-y coupling.

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BBU simulation

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  1. BBU simulation William Lou Supervisor: Prof. Georg Hoffstaetter

  2. BBU overview BBU theory Bmad simulation 2) CBETA BBU results What is CBETA HOM assignment statistics (1-pass and 4-pass) Aim for higher Results with additional phase-advances Results with x-y coupling Outline

  3. Beam breakup Instability (BBU) • Higher Order Modes (HOM) in the cavities give undesired kick. • Off-orbit bunch returns to the cavities and excite more HOMs… • BBU limits the maximum achievable current in an ERL threshold current • decreases with # recirculation turns.

  4. BBU theory: the mathematics Elementary case: 1-dipole-HOM + 1 recirculation pass Current Transverse offset Wake function [N/C^2] Transverse momentum gained from HOM Wake function (Cavity HOM property) An HOM is characterized by ( , , )

  5. HOM voltage T12 of the transfer matrix is a lattice property Transverse offset after recirculation Measured current Bunch time spacing Current as a train of (dirac) beam bunches

  6. To solve this difference equation, we write HOM voltage (retaining all possible frequencies ) as: After some math, we obtain the dispersion relation between and At a given , multiple (complex) can satisfy the relation. At a stable , all must have negative imaginary part. At the threshold current, one crosses the real axis!!

  7. Find the critical which gives the maximum real 1-pass 1-HOM thin-lens cavity (1) General analytic formula The corresponding is

  8. BBU theory

  9. BBU theory

  10. BMAD software • Developed by Cornell LEPP • Open source, free, compiled in C and Fortran • Multi-pass lattice design, lattice optimization, multi-particle tracking algorithms, wakefields, Taylor maps, real-time control “knobs”… • https://www.classe.cornell.edu/~dcs/bmad/overview.html

  11. Theory v.s simulation

  12. Theory v.s simulation DR scan for 1 dipole-HOM 2-pass (Np = 4) lattice

  13. Find the critical which gives the maximum real eigenvalue of M BBU theory General formula (Np-pass, N-cav) for The corresponding is

  14. BBU simulation on Bmad • Given a complete lattice with multi-pass cavities and HOMs assigned… • Starts with a test current… • tracks off-orbit bunches through lattice • computes bunch-HOM momentum exchanges • determines stability of all HOM voltages • Attempts different test currents to pin down

  15. CBETA

  16. Simulated HOM data for one CBETA cavity • Cavity construction error: ± 125 μm, ± 250 μm… • 400 unique cavities provided per error case. The “10 worst dipole HOMs” (large figure of merit) provided per cavity.

  17. CBETA 1-pass Cavity shape error: 125 μm 499 out of 500 results above 40mA

  18. CBETA 4-pass Cavity shape error: 125 μm 494 out of 500 results below 40mA

  19. CBETA 4-pass With various cavity shape errors 125 μm 250 μm 500 μm 1000 μm

  20. Summary on Statistics • For the current CBETA design, the • can usually reach the high goal of 40 mA. • Cavity shape matters!

  21. Aim for better • Potential ways to improve : 1) Change bunch frequency 2) Introduce additional phase advance 3) Introduce x-y coupling

  22. Does vary with bunch frequency?

  23. Vary the optics in Bmad Two cases: Introduce either T matrix at the end of LINAC 1st pass

  24. CBETA 1-pass v.s additional phase advances(decoupled optics) Min = 140 mA Max = 611 mA nominal = 342 mA results can improve significantly

  25. 1-pass decoupled with different HOM assignments… Min = 175 mA Max = 640 mA Nominal = 218 mA Min = 165 mA Max = 595 mA Nominal = 248 mA

  26. CBETA 1-pass with x-y coupling Min = 140 mA Max = 520 mA nominal = 342 mA results can improve significantly

  27. 1-pass coupled with different HOM assignments… Min = 136 mA Max = 436 mA Nominal = 218 mA Min = 166 mA Max = 678 mA Nominal = 248 mA

  28. CBETA 4-pass v.s additional phase advances(decoupled optics) Min = 61 mA Max = 193 mA Nominal = 69 mA results can improve

  29. 4-pass decoupled with different HOM assignments… Min = 37 mA Max = 176 mA Nominal = 38 mA Min = 39 mA Max = 205 mA Nominal = 49 mA

  30. CBETA 4-pass with x-y coupling Min = 89 mA Max = 131 mA Nominal = 69 mA results can improve

  31. 4-pass coupled with different HOM assignments… Min = 47 mA Max = 100 mA Nominal = 38 mA Min = 57 mA Max = 107 mA Nominal = 49 mA

  32. For both CBETA 1-pass and 4-pass, introducing additional phase advances is more promising than x-y coupling • In reality the phase variations are restricted by other optical constraints

  33. Summary • Bmad simulation agrees well with BBU theory • For both CBETA 1-pass and 4-pass , 98% simulated are above the high goal (40mA) • Introducing additional phase advances gives promise to improve the for CBETA

  34. Acknowledgment Special thanks to: • Prof. Georg Hoffstaetter • Christopher Mayes • Nick Valles • David Sagan References • BBU papers by Georg and Ivan • CBETA review meeting documents • Bmad manual • Nick Valles’s dissertation

  35. THE END

  36. RFC cavity HOMs from Nick Valles Helper Slide #2

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