1. 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

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  transition, crossing (PS, RHIC) or isochronous rings ( -factory proton driver accumulator) E.Benedetto, GSI collaboration meeting 19-2-09

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

6. The two codes 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 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>>I th Courtesy B. Salvant

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

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

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

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

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

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

14. HEADTAIL Instability risetime computed by exponential fit over the horizontal centroid amplitude growth: 1e-3<x<10m Nb=0.2 10 12 E.Benedetto, GSI collaboration meeting 19-2-09

15. HEADTAIL  does not depend on Q s  is inversely proportional to N b N b =0.2 10 12 N b =1.0 10 12 parameters SPS beam @ 26GeV BB resonator: 1GHz 10 M  /m Q=1 I th =0.5mA N b,th =~7.2 10 10 Q s =10 -3 synchr motion OFF Q s =10 -3 synchr motion OFF E.Benedetto, GSI collaboration meeting 19-2-09

16. HEADTAIL (  x N b ) 1 kick/turn 10 kicks/turn 100 kicks/turn  s  1 kick/turn 10 kicks/turn 100 kicks/turn parameters SPS beam @ 26GeV BB resonator: 1GHz 10 M  /m Q=1 I th =0.5mA N b,th =~7.2 10 10 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 in very good agreement Far above threshshold a simple formula, (TMC model with only the 2 most critical modes) gives good approx: –  independent of Ts (as expected) –  proportional to 1/I b The comparison was made for SPS “short” bunches. What happens for “long” bunches (PS, -factory proton driver, …)? for TMCIdoesn’t know TMCI E.Benedetto, GSI collaboration meeting 19-2-09