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THE BATMAN INSTABILITY OF THE HERA PROTON RING

THE BATMAN INSTABILITY OF THE HERA PROTON RING. E . Metral (CERN/AB-ABP-HEI). With F. Willeke and G. Hoffstaetter (DESY). Introduction Observations Theory of coupled Landau damping Comparison between theory and experiments Conclusion. INTRODUCTION.

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THE BATMAN INSTABILITY OF THE HERA PROTON RING

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  1. THE BATMAN INSTABILITY OF THE HERA PROTON RING E. Metral (CERN/AB-ABP-HEI) With F. Willekeand G. Hoffstaetter (DESY) • Introduction • Observations • Theory of coupled Landau damping • Comparison between theory and experiments • Conclusion

  2. INTRODUCTION • Since the first start-up of HERA in 1992, a transverse coherent instability has appeared from time to time at the beginning of the acceleration ramp • In this process, the emittance is blown up and the beam is partially or completely lost • Although the instability was found to be of the head-tail type, and the chromaticity and linear coupling between the transverse planes was recognized as essential for the instability to occur, the driving mechanism was never clarified • A possible explanation of the phenomenon is presented using the coupled Landau damping theory

  3. OBSERVATIONS (1/3) • The characteristics behaviour of a head-tail instability m=0 has been clearly evidenced This means that at least one of the transverse chromaticities is becoming negative, since the initial controlled chromaticities are slightly positive, due to the fact that the machine is working above transition, to prevent the head-tail mode m=0 from developing This may be due to the persistent currents in the super-conducting magnets, which increase one chromaticity and decrease the other one. Furthermore, these persistent currents also introduce strong linear coupling between the transverse planes, which is identified by the tune distance on the coupling resonance

  4. OBSERVATIONS (2/3) • After careful investigations made with controlled linear coupling and chromaticity changes on the injection flat-bottom at 40 GeV, the following conclusions can be drawn : • Some coupling between the transverse planes is necessary for the instability to start • It is mainly a single-bunch effect, although extra multi-bunch effects can also be present • Both transverse planes are affected, due to coupling • There is a clear chromaticity dependence, the instability appearing for small negative values of the chromaticity • The most critical value of the chromaticities is , but it can also appear for positive chromaticities

  5. OBSERVATIONS (3/3) • Growth times of the order of 100 to 700 ms were measured with a single bunch and • If the chromaticities are set to , and the particles at large amplitudes are cut by collimators partially closed, the beam is clearly unstable • If is set to a large value, the beam is stabilised • To suppress the instability, the chromaticities are set to and linear coupling is compensated to have some safety margins

  6. THEORY OF COUPLED LANDAU DAMPING (1/7) • The theory of coupled Landau damping reveals the possibility to stabilise a beam by sharing the transverse instability growth rates and the stabilising tune spreads, which are responsible for Landau damping • A particular case (to clearly see the effect of linear coupling on Landau damping) : No horizontal tune spread and no vertical wake field • One-dimensional (Keil-Zotter) stability criteria (elliptical spectra)  H-plane is unstable  V-plane is stable

  7. THEORY OF COUPLED LANDAU DAMPING (2/7) • In the presence of linear coupling Skew quads strength Stable region

  8. THEORY OF COUPLED LANDAU DAMPING (3/7) • Conclusion : • This result demonstrates the sharing of Landau damping from the stable vertical to the unstable horizontal plane, relating the vertical spread to the horizontal impedance • It reveals another important feature of the coupling : The possible cancellation of the effect of the real frequency shift • Below a certain coupling strength the beam is still unstable: There is not enough Landau damping transferred to the unstable plane • Above a certain coupling strength the beam is again unstable: The coherent frequencies fall outside the incoherent frequency spreads and Landau damping can’t exist •  Beam stability can be obtained for intermediate values of skew quadrupole strength

  9. THEORY OF COUPLED LANDAU DAMPING (4/7) OBSERVATIONS OF THE BENEFICIAL EFFECT OF LINEAR COUPLING • CERN-PS  Beam for LHC • 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” • 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”

  10. THEORY OF COUPLED LANDAU DAMPING (5/7) • But, as seen before the theory also reveals the possibility to destabilise a beam in the case of a too strong coupling. In this case, the coherent tune is shifted outside the incoherent spectrum. The Landau damping mechanism is suppressed and the beam becomes unstable • This is the phenomenon, which is suspected to happen in the HERA proton ring

  11. THEORY OF COUPLED LANDAU DAMPING (6/7) • Considering the case, where the tune distributions and complex coherent tune shifts are the same in both transverse planes, the stability criterion is given by Sacherer formula Half Width at the Bottom of an elliptical spectrum It is the normal mode tune difference on the coupling resonance = Closest tune approach

  12. THEORY OF COUPLED LANDAU DAMPING (7/7) • If , i.e. in the absence of linear coupling, the one-dimensional (Keil-Zotter) stability criterion is recovered • If increases, the betatron frequency spread has to be increased Any coupling is bad in that particular case since the stability condition is more restrictive than the one-dimensional stability criterion • When and , the formula reduces to the following simple stability criterion

  13. COMPARISON BETWEEN THEORY AND EXPERIMENTS (1/8) • Machine and beam parameters  Close to the coupling resonance

  14. COMPARISON BETWEEN THEORY AND EXPERIMENTS (2/8) • The coupling impedance, which is suspected to drive the instability, is the resistive-wall impedance. In the thick-wall approximation, i.e. when the skin depth is smaller than the wall thickness, it is given by Neglecting theinductive bypass ! Vacuum chamber resistivity (for the warm stainless steel, which has the dominant effect) The warm sections occupy 15% of the circumference and

  15. COMPARISON BETWEEN THEORY AND EXPERIMENTS (3/8) Transverse power spectrum for the head-tail modes (solid lines) and resistive part of the resistive-wall impedance (dotted lines)

  16. COMPARISON BETWEEN THEORY AND EXPERIMENTS (4/8) • Theoretical results • In the absence of both linear coupling and Landau damping, the rise-time is  Therefore, in the absence of any stabilising mechanism, a head-tail instability of mode m=0 should develop with a rise-time of about 400 ms, which is in agreement with the observations( 4) , 6) and 8) )

  17. COMPARISON BETWEEN THEORY AND EXPERIMENTS (5/8) Rise-time of ~ 440 ms for few seconds. Both transverse planes are affected, due to coupling Growth time acquisition: The transverse position signals of the bunch center of charge from horizontal and vertical pick-ups are displayed as a function of time. Time scale: 500 ms/div.

  18. COMPARISON BETWEEN THEORY AND EXPERIMENTS (6/8) • The most critical value of the chromaticity found experimentally ( 5) ), is also the most critical one theoretically, if one considers the resistive-wall impedance • The resistive-wall impedance also explains the fact that it is mainly a single-bunch effect, but multi-bunch effects can be present as well ( 2) ) • Applying Keil-Zotter stability criterion in the absence of coupling, the necessary tune transverse spreads to stabilise the head-tail mode m=0 by Landau damping are given by

  19. COMPARISON BETWEEN THEORY AND EXPERIMENTS (7/8) Stability is thus ensured if such (small) tune spreads are present in the beam due to magnetic non-linearities • It is known that this is the case, since when the particles at large amplitudes are cut by collimators partially closed, the beam is clearly unstable ( 7) ) • In the presence of a sufficiently small coupling (i.e. when the stability criterion is fulfilled),the beam should be stable, even with the most critical value of the chromaticity  In agreement with the observations ( 1) ) • In the presence of too strong coupling, i.e. when the stability criterion is not fulfilled, the beam becomes unstable and both planes are affected due to coupling,in agreement with the observations ( 3) )

  20. COMPARISON BETWEEN THEORY AND EXPERIMENTS (8/8) • The beamis stable ifis decreased and is increased (to move away from the coupling resonance  Decouple the machine) As observed, but and is the “magic working point” • To have some margins during the acceleration ramp where all the parameters are not perfectly controlled, the chromaticities should also be set to positive values, e.g. , since the high-order modes are more difficult to drive  It is also in agreement with the observations ( 9) )

  21. CONCLUSION (1/3) • All the measurements performed so far on the so-called “Batman instability” in the HERA proton ring, can be explained by the theory of coupled Landau damping of a head-tail instability due to the resistive-wall impedance  New name suggested : “coupled head-tail instability” • In the absence of coupling, this instability is stabilised by Landau damping in both planes whatever the chromaticity is • When coupling becomes too strong, then the coherent tune is shifted outside the incoherent spectrum. The Landau damping mechanism is suppressed and the beam is unstable • To prevent the instability from developing, the normal mode tune difference on the coupling resonance has to be kept smaller than the half width at the bottom of the tune distribution

  22. CONCLUSION (2/3) • What about the inductive bypass, which has been neglected in this analysis ? Rise-Time [s]of the mode m = 0 • Without inductive bypass • With inductive bypass  Same results with or without inductive bypass

  23. CONCLUSION (3/3) Reminder : Non inductive-bypass regime  and the first unstable betatron line is at ~ 30 kHz …

  24. ACKNOWLEDGEMENTS Many thanks to F. Willeke for helpful discussions and for providing the possibility of this collaboration

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