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E. Benedetto, E. Metral Acknowledgements: G. Rumolo, D. Quatraro, B. Salvant (CERN) 19/2/09

Instability rise-time far above the TMCI threshold: Comparison between simple theory, MOSES and HEADTAIL. E. Benedetto, E. Metral Acknowledgements: G. Rumolo, D. Quatraro, B. Salvant (CERN) 19/2/09. CERN/GSI beam dynamics and collective effects collaboration meeting. Outline. Motivation

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E. Benedetto, E. Metral Acknowledgements: G. Rumolo, D. Quatraro, B. Salvant (CERN) 19/2/09

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  1. Instability rise-time far above the TMCIthreshold: Comparison between simple theory, MOSES and HEADTAIL E. Benedetto, E. Metral Acknowledgements: G. Rumolo, D. Quatraro, B. Salvant (CERN) 19/2/09 CERN/GSI beam dynamics and collective effects collaboration meeting

  2. Outline • Motivation • TMC theory to compute rise-time far above threshold • Simple TMC model, MOSES, HEADTAIL: • Qualitative • Quantitative • Conclusions and discussion E.Benedetto, GSI collaboration meeting 19-2-09

  3. Transverse Instability for high-intensity single-bunch beams • In the past, studies have been done for what concerns finding the instability threshold • Different approaches: • Beam Break-up • TMC theory • Coasting beam with peak value • post Head-Tail • fast blow-up Unified the different approaches and formalisms to compute instability threshold → E.Metral, 2004 E.Benedetto, GSI collaboration meeting 19-2-09

  4. Transverse Instability for high-intensity single-bunch beams • Next step: • for intensities far above the TMCI intensity threshold • i.e. instability risetime much faster then synchrotron period • How to evaluate the risetime? Can we still use the concept of modes and modes coupling? → Follow-up discussion with W. Fisher and G. Rumolo at the CARE-HHH workshop (24-25/11/08, Chavannes-de-Bogis) → E.Metral, LIS meeting 1/12/08, https://ab-dep-abp.web.cern.ch/ab-dep-abp/LIS/Minutes/2008/20081201/metral1.pdf • Interesting for instance near g transition, crossing (PS, RHIC) or isochronous rings (n-factory proton driver accumulator) E.Benedetto, GSI collaboration meeting 19-2-09

  5. TMC theory and intensity threshold • Comparison HEADTAIL vs. MOSES approaching Ith • Very good agreement between the 2 codes for what concerns mode shifts and instability threshold • parameters SPS beam @ • 26GeV • BB resonator: • 1GHz • 10 MW/m • Q=1 E. Metral, B. Salvant, G. Rumolo, … Ith=0.5mA Nb~7.2 1010 The instability seen by HEADTAIL is therefore clearly a TMCI! E.Benedetto, GSI collaboration meeting 19-2-09

  6. MOSES (Y.H. Chin, CERN-LEP-Div-Rep-88-005-TH) It solves Sacherer integrals Mode shifts and coupling due to the interaction of a bunch with an impedance (BB resonator) It has been developed for the TMCI HEADTAIL (G. Rumolo, F. Zimmermann, SL-Note 2002-036-AP, CERN 2002) Macroparticle simulations, the bunch is sliced and interacts slice-by-slice with the wake-fields. Doesn’t know anything about TMCI or modes Localized impedance source The two codes Courtesy G.Rumolo E.Benedetto, GSI collaboration meeting 19-2-09

  7. TMC theory and intensity threshold • Extension of TMCI theory far above TMCI threshold • Comparison theory - HEADTAIL – MOSES for I>>Ith Courtesy B. Salvant

  8. Linear Nonlinear Infinite rise-time MOSES • Imaginary part of the modes shift vs. Ib • Risetime E.Benedetto, GSI collaboration meeting 19-2-09 E.Metral, LIS meeting 1/12/08

  9. MOSES • parameters SPS beam @ • 26GeV • BB resonator: • 1GHz • 10 MW/m • Q=1 E.Benedetto, GSI collaboration meeting 19-2-09 E.Metral, LIS meeting 1/12/08

  10. MOSES • parameters SPS beam @ • 26GeV • BB resonator: • 1GHz • 10 MW/m • Q=1 E.Benedetto, GSI collaboration meeting 19-2-09 E.Metral, LIS meeting 1/12/08

  11. MOSES • parameters SPS beam @ • 26GeV • BB resonator: • 1GHz • 10 MW/m • Q=1 E.Benedetto, GSI collaboration meeting 19-2-09 E.Metral, LIS meeting 1/12/08

  12. MOSES + • parameters SPS beam @ • 26GeV • BB resonator: • 1GHz • 10 MW/m • Q=1 E.Benedetto, GSI collaboration meeting 19-2-09 E.Metral, LIS meeting 1/12/08

  13. Simple TMC model with the 2 most critical modes with  is independent of synchrotron motion as could be anticipated (as the instability rise-time is much faster than synchrotron period) Furthermore  E.Benedetto, GSI collaboration meeting 19-2-09 E.Metral, LIS meeting 1/12/08

  14. HEADTAIL • Instability risetime computed by exponential fit over the horizontal centroid amplitude growth: Nb=0.2 1012 Nb=0.2 1012 1e-3<x<10m E.Benedetto, GSI collaboration meeting 19-2-09

  15. Nb=0.2 1012 Nb=1.0 1012 • parameters SPS beam @ • 26GeV • BB resonator: • 1GHz • 10 MW/m • Q=1 • Ith=0.5mA • Nb,th=~7.2 1010 HEADTAIL • t does not depend on Qs • t is inversely proportional to Nb Qs=10-3 synchr motion OFF Qs=10-3 synchr motion OFF E.Benedetto, GSI collaboration meeting 19-2-09

  16. parameters SPS beam @ • 26GeV • BB resonator: • 1GHz • 10 MW/m • Q=1 • Ith=0.5mA • Nb,th=~7.2 1010 HEADTAIL t (ms) (t x Nb) 1 kick/turn 10 kicks/turn 100 kicks/turn 1 kick/turn 10 kicks/turn 100 kicks/turn E.Benedetto, GSI collaboration meeting 19-2-09

  17. Some numerical values • Let’s consider I=100mA • MOSES: • Simple TMC model (2 most critical modes) • HEADTAIL: E.Benedetto, GSI collaboration meeting 19-2-09

  18. Conclusion • Answer to the question (of W.Fisher and others) is: Yes! We can still use the concept of modes and modes coupling to deduce the rise-time far above threshold… …since MOSES and HEADTAIL are invery good agreement • Far above threshshold a simple formula, (TMC model with only the 2 most critical modes) gives good approx: • tindependent of Ts (as expected) • tproportional to 1/Ib • The comparison was made for SPS “short” bunches. What happens for “long” bunches (PS, n-factory proton driver, …)? for TMCI doesn’t know TMCI E.Benedetto, GSI collaboration meeting 19-2-09

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