1. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 1/15 TRANSVERSE LANDAU DAMPING WITH SPACE CHARGE u Introduction and motivation u Review of our 2004 paper  Should give a 1 st 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 u Conclusions and future work Elias Métral (~10 min, 15 slides) Elias Métral (~10 min, 15 slides) With F. Ruggiero, CERN-AB-2004-025 (ABP)

2. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 2/15 u 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 u In 1995 Möhl extended these results to head-tail modes in bunched beams u 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 INTRODUCTION AND MOTIVATION (1/2)

3. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 3/15 u 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 u This issue is important for the LHC at injection INTRODUCTION AND MOTIVATION (2/2)

4. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 4/15 u Consider first the case of a (initially stable) coasting beam u 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 REVIEW OF OUR 2004 PAPER (1/10)

5. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 5/15 u Ex.1: Case of the space charge impedance (round beam in a round pipe) REVIEW OF OUR 2004 PAPER (2/10) u Definition of the transverse impedance u Ex.2: Case of a flat vacuum chamber From the detuning (or quadrupolar  incoh.) wake  This explains why a zero horizontal coherent tune shift is measured in flat chambers (as e.g. in the SPS)

6. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 6/15 u 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 REVIEW OF OUR 2004 PAPER (3/10) u Comparison with the case without space charge and for rigid dipole oscillations (m = 0) from Berg-Ruggiero1996

7. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 7/15 u Case of a quasi-parabolic (n = 2) distribution function having the same normalized rms beam size in both transverse planes REVIEW OF OUR 2004 PAPER (4/10) for and with OctupolesSpace charge n = 2

8. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 8/15 u Transverse beam profiles REVIEW OF OUR 2004 PAPER (5/10) Transverse beam profiles LHC collimators setting

9. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 9/15 u Space-charge force REVIEW OF OUR 2004 PAPER (6/10)

10. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 10/15 u (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 REVIEW OF OUR 2004 PAPER (7/10) with It is 4 in the case of a bi-Gaussian

11. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 11/15 u 2D tune footprint REVIEW OF OUR 2004 PAPER (8/10) Real Approximation with linear terms only (adapting the coefficients!) Low-intensity working point

12. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 12/15 u 3D view of the 2D tune footprint REVIEW OF OUR 2004 PAPER (9/10) Low-intensity working point Real Approximation

13. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 13/15 REVIEW OF OUR 2004 PAPER (10/10) Stability diagrams for nominal LHC parameters at injection and maximum permitted octupolar strength Height given by the octupoles 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 u Analytical stability diagrams derived in the approximate case Stable point with octupoles alone but unstable when SC added

14. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 14/15 CONCLUSION AND FUTURE WORK (1/2) u The previous results should give a reasonable picture for coasting beams or bunches with rectangular longitudinal profiles u 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 u In the absence of space charge, the stability diagrams of Berg- Ruggiero1996 are also recovered u These results should give a 1 st 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. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 15/15 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. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 16/15 APPENDIX 1: Evolution of the stability diagrams with decreasing intensity (1/4)

17. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 17/15 APPENDIX 1: Evolution of the stability diagrams with decreasing intensity (2/4)

18. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 18/15 APPENDIX 1: Evolution of the stability diagrams with decreasing intensity (3/4)

19. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 19/15 APPENDIX 1: Evolution of the stability diagrams with decreasing intensity (4/4)

20. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 20/15 APPENDIX 2: Evolution of the stability diagrams with decreasing octupolar strength (1/4)

21. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 21/15 APPENDIX 2: Evolution of the stability diagrams with decreasing octupolar strength (2/4)

22. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 22/15 Change of vertical scale ! APPENDIX 2: Evolution of the stability diagrams with decreasing octupolar strength (3/4)

23. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 23/15 APPENDIX 2: Evolution of the stability diagrams with decreasing octupolar strength (4/4) Change of vertical scale !

24. Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06 24/15 APPENDIX 3: PS experiment u 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 u 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