1. Elias Metral, CERN-PS seminar, 12/04/20001 u Introduction, observations and motivation u Theory u Experiments u Conclusion STABILISING INTENSE BEAMS BY LINEAR COUPLING Elias METRAL

2. Elias Metral, CERN-PS seminar, 12/04/20002 INTRODUCTION u Low intensity  Single-particle phenomena u High intensity  Collective effects 2 stabilising mechanisms against transverse coherent instabilities : n Landau damping by non-linearities (space-charge and octupoles) Non-linearities  Perturbations of the single-particle motion (resonances) Single-particle trajectory Circular design orbit ! One particle n Feedback systems

3. Elias Metral, CERN-PS seminar, 12/04/20003 OBSERVATIONS u In 1989, a coherent instability of the quadrupolar mode type driven by ions from the residual gas has been observed by D. Mohl et al. in the CERN-AA and successfully cured by adjusting both tunes close to 2.25 u In 1993, a single-bunch instability of the dipolar mode type driven by the resistive wall impedance has been observed by R. Cappi in the CERN-PS and “sometimes cured” by adjusting both tunes close to 6.24 THE IDEA (from R. CAPPI and D. MOHL) WAS TO : USE LINEAR COUPLING TO “TRANSFER DAMPING” FROM THE STABLE TO THE UNSTABLE PLANE, IN ORDER TO REDUCE THE EXTERNAL NON-LINEARITIES

4. Elias Metral, CERN-PS seminar, 12/04/20004 THEORY (1/16) u A general formula for the transverse coherent instabilities with n Frequency spreads (due to octupoles) n Linear coupling (due to skew quadrupoles) x-dispersion integral x-Sacherer’s formula Mode coupling term Linear coupling term

5. Elias Metral, CERN-PS seminar, 12/04/20005 THEORY (2/16) Near the coupling resonance Uncorrelated distribution functions (Averaging method) is the lth Fourier coefficient of the normalized skew gradient Coherent frequency to be determined

6. Elias Metral, CERN-PS seminar, 12/04/20006 THEORY (3/16) Sacherer’s formula (single- and coupled-bunch instabilities) => “low intensity” case Head-tail modes Coupled-bunch modes Power spectrumPick-up (Beam Position Monitor) signal One particular turn Time  -signal Time

7. Elias Metral, CERN-PS seminar, 12/04/20007 THEORY (4/16) n In the absence of- Linear coupling - Mode coupling u Let’s recover the 1D results => Sacherer’s formula is recovered Instability Motions => Instability growth rate Real coherent betatron frequency shift l In the absence of frequency spreads These are the Laslett, Neil and Sessler (LNS) coefficients for coasting beams

8. Elias Metral, CERN-PS seminar, 12/04/20008 THEORY (5/16) l In the presence of frequency spreads (1) Lorentzian distribution 1D criterion (2) Elliptical distribution 1D criterion Keil-Zotter’s stability criterion Overestimates Landau damping (infinite tails) Underestimates Landau damping (sharp edges)

9. Elias Metral, CERN-PS seminar, 12/04/20009 THEORY (6/16) n In the absence of linear coupling but in the presence of mode coupling => “high intensity” case => Kohaupt’s stability criterion against Transverse Mode Coupling Instability (TMCI) is recovered l In the absence of frequency spreads l In the presence of frequency spreads => A tune spread of the order of the synchrotron tune is needed for stabilisation by Landau damping

10. Elias Metral, CERN-PS seminar, 12/04/200010 THEORY (7/16) u New 2D results n In the absence of mode coupling only l In the absence of frequency spreads Necessary condition for stability Transfer of growth rates Stable region No coupling Full coupling Stability criteria : Full coupling? Stability criterion (for each mode m) for coupled-bunch modes (and coasting beams)

11. Elias Metral, CERN-PS seminar, 12/04/200011 THEORY (8/16) => Normalised coupling (or sharing) function for full coupling

12. Elias Metral, CERN-PS seminar, 12/04/200012 THEORY (9/16) l In the presence of frequency spreads (1) Lorentzian distribution => Same results with replaced by No coupling Full coupling Stability criteria : Transfer of both instability growth rates and frequency spreads (Landau damping)

13. Elias Metral, CERN-PS seminar, 12/04/200013 THEORY (10/16) (2) Elliptical distribution o A particular case : No horizontal tune spread and no vertical wake field Stable region

14. Elias Metral, CERN-PS seminar, 12/04/200014 THEORY (11/16) 1) “far from” 2) “near” THE TUNE SEPARATION SHOULD BE SMALLER THAN THE ORDER OF MAGNITUDE OF IN ORDER TO HAVE THE TRANSFER OF LANDAU DAMPING o Approximate general stability criterion => Transfer of growth rates only Necessary condition

15. Elias Metral, CERN-PS seminar, 12/04/200015 THEORY (12/16) H-plane V-plane Transfer of frequency spread (to Landau damp ) Same result obtained considering both non-linear space-charge forces and octupoles for coasting beams => D. Mohl and H. Schonauer’s 1D stability criterion (gain of factor ~2) On the coupling resonance  “One plane is stabilised by Landau damping and the other one is stabilised by coupling”

16. Elias Metral, CERN-PS seminar, 12/04/200016 THEORY (13/16) n In the presence of both mode coupling and linear coupling, neglecting frequency spreads => Necessary condition for stability

17. Elias Metral, CERN-PS seminar, 12/04/200017 THEORY (14/16) => Computed gain in intensity of about 50% for the classical ratio of factor 2 between the transverse sizes of the vacuum chamber Example :

18. Elias Metral, CERN-PS seminar, 12/04/200018 THEORY (15/16) SHARING OF DAMPING BY FEEDBACKS The stabilising effect of feedbacks can be introduced in the coefficient An electronic feedback system can be used to damp transverse coherent instabilities. Its action on the beam can be described in terms of an impedance, which depends on the distance between pick-up and kicker, and the electronic gain and time delays Kicker Electronics Beam  Its damping effect in one plane, can also be transferred to the other plane using coupling Pick-up

19. Elias Metral, CERN-PS seminar, 12/04/200019 THEORY (16/16) SUMMARY OF THEORY u 1 general formula for transverse coherent instabilities in the presence of n Frequency spreads (due to octupoles) n Linear coupling (due to skew quadrupoles) u In the absence of coupling the well-known 1D results are recovered as expected u Effects of linear coupling (skew quadrupoles and/or tune distance from coupling resonances) : n Transfer of growth rates for “any” coupling n Transfer of Landau damping for “optimum” coupling “Chromaticity sharing” (for Sacherer’s formula) Linear coupling is an additional (3 rd ) method that can be used to damp transverse coherent instabilities =>

20. Elias Metral, CERN-PS seminar, 12/04/200020 u Experimental conditions n High intensity bunched proton beam n 1.2 s long flat bottom at injection kinetic energy EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (1/9) u Sacherer’s formula => coupled-bunch instabilities n Coupled-bunch modes n Most critical head-tail mode number for the horizontal plane  Landau damping is needed 121 s -1 - 40 s -1

21. Elias Metral, CERN-PS seminar, 12/04/200021 u Observations 10 dB/div SWP 1.2 s  R signal Spectrum Analyzer (zero frequency span) Beam-Position Monitor (20 revolutions superimposed) One particular turn Center 360 kHz RES BW 10 kHz VBW 3 kHz Time EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (2/9) 1D case  See next slides Time (20 ns/div)

22. Elias Metral, CERN-PS seminar, 12/04/200022 MEASUREMENT OF THE CERN-PS LINEAR COUPLING u In the presence of linear coupling between the transverse planes, the difference from the tunes of the 2 normal modes is given by u Measurement method : For different skew quadrupole currents, we increase and decrease in the vicinity of the coupling resonance and we measure the 2 normal mode frequencies using a vertical kicker, a vertical pick-up and a FFT analyzer u In the PS n Coupling resonance n No solenoid Guignard’s coupling coefficient It is obtained from the general formula (in the smooth approximation used to study instabilities) EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (3/9)

23. Elias Metral, CERN-PS seminar, 12/04/200023 u Coupling measurements from mode frequencies by FFT analysis n Low intensity bunched proton beam n 1.2 s long flat bottom at injection kinetic energy u “Mountain range” display for the “natural” coupling Frequency Time FFT Analyzer EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (4/9)

24. Elias Metral, CERN-PS seminar, 12/04/200024 => Modulus of the normalised skew gradient vs. skew quadrupole current EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (5/9)

25. Elias Metral, CERN-PS seminar, 12/04/200025 u Stabilisation by Landau damping (1D case) n Theoretical frequency spread required This is less than required by the theory by a factor 3 (without taking into account space-charge non-linearities...) n Simplified (elliptical) stability criterion : Keil-Zotter’s criterion n Experimental frequency spread required  EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (6/9)

26. Elias Metral, CERN-PS seminar, 12/04/200026 u Stabilisation by coupled Landau damping (2D) Measurement Theory (Lorentzian vertical distribution) n Constant tune separation EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (7/9)

27. Elias Metral, CERN-PS seminar, 12/04/200027 Measurement Theory (Lorentzian vertical distribution) n Constant octupole strength EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (8/9)

28. Elias Metral, CERN-PS seminar, 12/04/200028 u The experimental results confirm the predicted beneficial effect of coupling on Landau damping u Using coupling, a factor 7 has been gained in the octupole current (for this particular case) => Less non-linearities u Difference between theoretical predictions and experiments  Space-charge non-linearities, impedance and tune spread models… u Further theoretical work => More precise treatment of the non- linearities in the normal modes CONCLUSIONS OF EXPERIMENT-1 EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (9/9)

29. Elias Metral, CERN-PS seminar, 12/04/200029 EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (1/6) u 1.2 s long flat bottom at injection kinetic energy u Bunch length u Transverse tunes u Transverse chromaticities Head-tail mode number m Growth rates [s -1 ] Sacherer’s formula => u Single bunch of protons with nominal intensity

30. Elias Metral, CERN-PS seminar, 12/04/200030 Time 10 dB/div SWP 1.2 s (20 ns/div)  R signal Spectrum Analyzer (zero frequency span) Beam-Position Monitor (20 revolutions superimposed) Center 355 kHz RES BW 10 kHz VBW 3 kHz Time EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (2/6) u Observations1D case 

31. Elias Metral, CERN-PS seminar, 12/04/200031 u Stabilisation by linear coupling only  ~ no emittance blow-up (limit) EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (3/6) since u The ~ same results are obtained for the ultimate beam  ~ no emittance blow-up but ~ no blow-up in the PS

32. Elias Metral, CERN-PS seminar, 12/04/200032 u Voir le file presentation 1 EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (4/6)

33. Elias Metral, CERN-PS seminar, 12/04/200033 u 8 bunches of protons with nominal intensity n Theoretical stabilising skew gradient  coupled-bunch instabilities  or EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (5/6) Head-tail mode number m Growth rates [s -1 ] u The ~ same results are obtained for the ultimate beam

34. Elias Metral, CERN-PS seminar, 12/04/200034 CONCLUSIONS OF EXPERIMENT-2 u The stability criterion for the damping of transverse head-tail instabilities in the presence of linear coupling only has been verified experimentally and compared to theory, leading to a good agreement (to within a factor smaller than 2) u The CERN-PS beam for LHC (nominal or ultimate intensity) CAN BE STABILISED using linear coupling only* (skew quadrupoles and/or tune separation). Furthermore, this result should be valid for “any” intensity (as concerns pure head- tail instabilities)... EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (6/6) * i.e. with neither octupoles nor feedbacks

35. Elias Metral, CERN-PS seminar, 12/04/200035 OBSERVATIONS OF THE BENEFICIAL EFFECT OF LINEAR COUPLING IN OTHER MACHINES u LANL-PSR (from B. Macek) “Operating at or near the coupling resonance with a skew quad is one of the most effective means to damp our 'e-p' instability” u BNL-AGS (from T. Roser) “The injection setup at AGS is a tradeoff between a 'highly coupled' situation, associated with slow loss, and a 'lightly coupled' situation where the beam is unstable (coupled-bunch instability)” u CERN-SPS (from G. Arduini) “A TMCI in the vertical plane with lepton beams at 16 GeV is observed. Using skew quads ('just turning the knobs'), gains in intensity of about 20-30%, and a more stable beam, have been obtained” => MDs are foreseen to examine these preliminary results in detail u CERN-LEP (from A. Verdier) “The TMCI in the vertical plane at 20 GeV sets the limit to the intensity per bunch. The operation people said that it's better to accumulate with tunes close to each other” => MDs are foreseen to examine these preliminary results in detail

36. Elias Metral, CERN-PS seminar, 12/04/200036CONCLUSION u These results explain why many high intensity accelerators and colliders work best close to a coupling resonance blablablabla and/or using skew quadrupoles. They can be used to find optimum values for the transverse tunes, the skew quadrupole and octupole currents, and the chromaticities (=> sextupoles) u The CERN-PS beam for LHC can be stabilised by linear coupling only u Linear coupling is also used at BNL and LANL, and seems to be helpful in SPS and LEP => See future MDs u Using this “simple” formalism, the following results are also obtained: n Coherent beam-beam modes => Decoupling the 2 beams by making the tune difference much larger than the beam-beam parameter (A. Hofmann) n 2-stream instabilities => Same stability criterion with negative coupling (Laslett, Mohl and Sessler) ACK. : R. CAPPI AND D. MOHL, M. MARTINI AND THE OPERATION STAFF THEIR IDEA !