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BBU simulation

Explore the theory of Beam Breakup Instability (BBU) and Higher Order Modes (HOM) in CBETA, analyzing results and strategies for increasing current efficiency using simulations on the Bmad software.

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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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