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Introduction, observations and motivation Theory Experiments Conclusion

STABILISING INTENSE BEAMS BY LINEAR COUPLING. Elias METRAL. Introduction, observations and motivation Theory Experiments Conclusion. INTRODUCTION. Single-particle trajectory. One particle. Circular design orbit. Low intensity  Single-particle phenomena

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Introduction, observations and motivation Theory Experiments Conclusion

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  1. STABILISING INTENSE BEAMS BY LINEAR COUPLING Elias METRAL • Introduction, observations and motivation • Theory • Experiments • Conclusion

  2. INTRODUCTION Single-particle trajectory One particle Circular design orbit • Low intensity  Single-particle phenomena • High intensity  Collective effects 2 stabilising mechanisms against transverse coherent instabilities : • Landau damping by non-linearities (space-charge and octupoles) Non-linearities  Perturbations of the single-particle motion (resonances) ! • Feedback systems

  3. OBSERVATIONS • 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 • 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. THEORY (1/16) • A general formula for the transverse coherent instabilities with • Frequency spreads (due to octupoles) • Linear coupling (due to skew quadrupoles) Mode coupling term Linear coupling term x-Sacherer’s formula x-dispersion integral

  5. THEORY (2/16) Uncorrelated distribution functions (Averaging method) Coherent frequency to be determined Near the coupling resonance is the lth Fourier coefficient of the normalized skew gradient

  6. THEORY (3/16) Sacherer’s formula (single- and coupled-bunch instabilities) => “low intensity” case Head-tail modes Coupled-bunch modes Power spectrum Pick-up (Beam Position Monitor) signal -signal -signal Time Time One particular turn

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

  8. THEORY (5/16) • In the presence of frequency spreads (1) Lorentzian distribution (2) Elliptical distribution Underestimates Landau damping (sharp edges) Overestimates Landau damping (infinite tails) 1D criterion 1D criterion Keil-Zotter’s stability criterion

  9. THEORY (6/16) • In the absence of linear coupling but in the presence of mode coupling => “high intensity” case • In the absence of frequency spreads => Kohaupt’s stability criterion against Transverse Mode Coupling Instability (TMCI) is recovered • In the presence of frequency spreads => A tune spread of the order of the synchrotron tune is needed for stabilisation by Landau damping

  10. Stability criterion (for each mode m) • New 2D results THEORY (7/16) • In the absence of mode coupling only for coupled-bunch modes (and coasting beams) • In the absence of frequency spreads Necessary condition for stability Transfer of growth rates Stability criteria : Stable region Full coupling? No coupling Full coupling

  11. THEORY (8/16) for full coupling => Normalised coupling (or sharing) function

  12. THEORY (9/16) • In the presence of frequency spreads (1) Lorentzian distribution => Same results with replaced by Stability criteria : No coupling Full coupling Transfer of both instability growth rates and frequency spreads (Landau damping)

  13. THEORY (10/16) (2) Elliptical distribution • A particular case : No horizontal tune spread and no vertical wake field Stable region

  14. THEORY (11/16) • Approximate general stability criterion => Transfer of growth rates only 1) “far from” 2) “near” Necessary condition THE TUNE SEPARATION SHOULD BE SMALLER THAN THE ORDER OF MAGNITUDE OF IN ORDER TO HAVE THE TRANSFER OF LANDAU DAMPING

  15. On the coupling resonance THEORY (12/16) H-plane Transfer of frequency spread (to Landau damp ) V-plane “One plane is stabilised by Landau damping and the other one is stabilised by coupling”  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)

  16. THEORY (13/16) • In the presence of both mode coupling and linear coupling, neglecting frequency spreads => Necessary condition for stability

  17. THEORY (14/16) Example : => Computed gain in intensity of about 50% for the classical ratio of factor 2 between the transverse sizes of the vacuum chamber

  18. THEORY (15/16) SHARING OF DAMPING BY FEEDBACKS 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 Electronics Pick-up Kicker Beam The stabilising effect of feedbacks can be introduced in the coefficient Its damping effect in one plane, can also be transferred to the other plane using coupling 

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

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

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

  22. EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (3/9) MEASUREMENT OF THE CERN-PS LINEAR COUPLING • Coupling resonance • No solenoid • In the PS • In the presence of linear coupling between the transverse planes, the difference from the tunes of the 2 normal modes is given by Guignard’s coupling coefficient It is obtained from the general formula (in the smooth approximation used to study instabilities) • 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

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

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

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

  26. EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (7/9) • Stabilisation by coupled Landau damping (2D) • Constant tune separation Measurement Theory (Lorentzian vertical distribution)

  27. EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (8/9) • Constant octupole strength Measurement Theory (Lorentzian vertical distribution)

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

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

  30. EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (2/6) • Observations 1D case  Spectrum Analyzer (zero frequency span) Beam-Position Monitor (20 revolutions superimposed) R signal 10 dB/div Center 355 kHz Time Time (20 ns/div) RES BW 10 kHz VBW 3 kHz SWP 1.2 s

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

  32. EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (4/6) • Voir le file presentation 1

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

  34. EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (6/6) CONCLUSIONS OF EXPERIMENT-2 • 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) • 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)... *i.e. with neither octupoles nor feedbacks

  35. OBSERVATIONS OF THE BENEFICIAL EFFECT OF LINEAR COUPLING IN OTHER MACHINES • 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” • 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-bunchinstability)” • 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 • 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. 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) • The CERN-PS beam for LHC can be stabilised by linear coupling only • Linear coupling is also used at BNL and LANL, and seems to be helpful in SPS and LEP => See future MDs • Using this “simple” formalism, the following results are also obtained: • Coherent beam-beam modes => Decoupling the 2 beams by making the tune difference much larger than the beam-beam parameter (A. Hofmann) • 2-stream instabilities => Same stability criterion with negative coupling (Laslett, Mohl and Sessler) CONCLUSION THEIR IDEA ! ACK. : R. CAPPI AND D. MOHL, M. MARTINI AND THE OPERATION STAFF

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