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Transverse Landau Damping with Space Charge

This presentation discusses the influence of space-charge non-linearities on the Landau damping mechanism of transverse coherent instabilities in high-intensity hadron beams. It reviews a 2004 paper and addresses questions about when a beam can become stable with added incoherent space-charge force and how a stable beam can become unstable only through the addition of incoherent space-charge force.

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Transverse Landau Damping with Space Charge

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  1. TRANSVERSE LANDAU DAMPING WITH SPACE CHARGE Elias Métral (~10 min, 15 slides) With F. Ruggiero, CERN-AB-2004-025 (ABP) • Introduction and motivation • Review of our 2004 paper  Should give a 1st answer to the questions: • When can a beam become stable by adding the direct (incoherent) space-charge force? • How can a stable beam become unstable (coherent motion) only by adding the direct (incoherent) space-charge force? • Why is the decoherence time much longer with space charge (as e.g. in the CERN PS)?  Same as before • Conclusions and future work

  2. INTRODUCTION AND MOTIVATION (1/2) • The influence of space-charge non-linearities on the Landau damping mechanism of transverse coherent instabilities has been studied in 1974 by Möhl and Schönauer for coasting and rigid bunched beams • In 1995 Möhl extended these results to head-tail modes in bunched beams • The basic results of these studies are that • In the absence of external (octupolar) non-linearities, the space-charge non-linearities have no effect on bean stability, as the incoherent space-charge tune spread moves with the beam • When octupoles are added, the incoherent space-charge tune spread is “mixed-in”, and in this case the octupole strength required for stabilization can depend strongly on the sign of the excitation current of the lenses

  3. INTRODUCTION AND MOTIVATION (2/2) • In the workshop on Instabilities of High Intensity Hadron Beams in Rings (BNL, 1999), the community was not entirely comfortable with the interpretation by Möhl and Schönauer • Uncertainty amongst some individuals in the meaning/interpretation of the transverse space-charge impedance (issue of coherent versus incoherent tune shift) • Recommendation for further controlled experiments • This issue is important for the LHC at injection

  4. REVIEW OF OUR 2004 PAPER (1/10) • Consider first the case of a (initially stable) coasting beam • 3 types of forces are taken into account • The external focusing force that depends on the horizontal deviation of the particle from a fixed reference (e.g. the centre of the chamber). The corresponding tune is • The coherent space-charge force that depends on the deviation of the beam centre from the centre of the chamber. The corresponding (generalised) tune shift is • The incoherent space-charge force that depends on the deviation of the particle from the beam centre. The corresponding (generalised) tune shift is

  5. REVIEW OF OUR 2004 PAPER (2/10) • Definition of the transverse impedance • Ex.1: Case of the space charge impedance (round beam in a round pipe) From the detuning (or quadrupolar  incoh.) wake • Ex.2: Case of a flat vacuum chamber  This explains why a zero horizontal coherent tune shift is measured in flat chambers (as e.g. in the SPS)

  6. REVIEW OF OUR 2004 PAPER (3/10) • In the case of a bunched beam (neglecting the longitudinal variation of the transverse space-charge force and the synchrotron tune spread), the 2-dimensional dispersion relation to be solved is given by • Comparison with the case without space charge and for rigid dipole oscillations (m = 0) from Berg-Ruggiero1996

  7. REVIEW OF OUR 2004 PAPER (4/10) • Case of a quasi-parabolic (n = 2) distribution function having the same normalized rms beam size in both transverse planes n = 2 for Octupoles Space charge and with

  8. REVIEW OF OUR 2004 PAPER (5/10) Transverse beam profiles • Transverse beam profiles LHC collimators setting

  9. REVIEW OF OUR 2004 PAPER (6/10) • Space-charge force

  10. REVIEW OF OUR 2004 PAPER (7/10) • (Non-linear) space-charge tune shift: The non-linear dependence of the force is converted into an amplitude dependence of the particle’s tune using the method of the harmonic balance, which is an averaging process over the incoherent betatron motions with It is 4 in the case of a bi-Gaussian

  11. REVIEW OF OUR 2004 PAPER (8/10) • 2D tune footprint Low-intensity working point Real Approximation with linear terms only (adapting the coefficients!)

  12. REVIEW OF OUR 2004 PAPER (9/10) • 3D view of the 2D tune footprint Real Approximation Low-intensity working point

  13. REVIEW OF OUR 2004 PAPER (10/10) • Analytical stability diagrams derived in the approximate case Stability diagrams for nominal LHC parameters at injection and maximum permitted octupolar strength Height given by the octupoles Stable point with octupoles alone but unstable when SC added Given by the small-amplitude space-charge tune shift Given by the large-amplitude space-charge tune shift + octupoles  Will move to the right with longitudinal motion

  14. CONCLUSION AND FUTURE WORK (1/2) • The previous results should give a reasonable picture for coasting beams or bunches with rectangular longitudinal profiles • In the absence of external (octupolar) non-linearities, the result of Möhl-Schönauer1974 is recovered: there is no stability region  Space-charge nonlinearities alone have no stabilizing effects • In the absence of space charge, the stability diagrams of Berg-Ruggiero1996 are also recovered • These results should give a 1st answer to the questions of the first page  The direct (incoherent) space-charge force can be responsible for beam instability, not by acting on the coherent tune but by shifting the incoherent tunes (leading to a loss of Landau damping)

  15. CONCLUSION AND FUTURE WORK (2/2) 2D tune footprint 3D tune footprint  The longitudinal variation of the transverse space-charge force (for the usual parabolic or Gaussian bunches) will increase the stable region on the right-hand side…

  16. APPENDIX 1: Evolution of the stability diagrams with decreasing intensity (1/4)

  17. APPENDIX 1: Evolution of the stability diagrams with decreasing intensity (2/4)

  18. APPENDIX 1: Evolution of the stability diagrams with decreasing intensity (3/4)

  19. APPENDIX 1: Evolution of the stability diagrams with decreasing intensity (4/4)

  20. APPENDIX 2: Evolution of the stability diagrams with decreasing octupolar strength (1/4)

  21. APPENDIX 2: Evolution of the stability diagrams with decreasing octupolar strength (2/4)

  22. APPENDIX 2: Evolution of the stability diagrams with decreasing octupolar strength (3/4) Change of vertical scale !

  23. APPENDIX 2: Evolution of the stability diagrams with decreasing octupolar strength (4/4) Change of vertical scale !

  24. APPENDIX 3: PS experiment • Measurements performed on a slow horizontal head-tail instability of high order (m = 6) due to the resistive-wall impedance developing on the long (1.2 s) injection flat-bottom in the presence of a large incoherent space-charge tune spread (~0.2), tend to confirm Möhl-Schönauer’s predictions • However, more detailed experiments are needed to better understand all the mechanisms involved, and in particular to distinguish between the cases where the coherent and incoherent tune shifts have the same or opposite sign, i.e. if the coherent tune is inside the incoherent space-charge tune spread or not

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