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STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES

STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES. E . Metral (CERN/AB-ABP-HEI). Introduction Dispersion relation Stability diagrams Parabolic distribution Gaussian distribution Distribution used by Sacherer for his stability criterion

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STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES

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  1. STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES E. Metral (CERN/AB-ABP-HEI) • Introduction • Dispersion relation • Stability diagrams • Parabolic distribution • Gaussian distribution • Distribution used by Sacherer for his stability criterion • Analytical computations for an elliptical spectrum • Application to the LHC at top energy

  2. INTRODUCTION • The most dangerous longitudinal single-bunch effect in the LHC is the possible suppression of Landau damping at top energy (7 TeV) • Consider the case of the LHC, i.e. an inductive impedance above transition It is said that “the coherent synchrotron frequency of the dipole mode does not move” Incoherent synchrotron frequency shift Incoh. spread How can the beam be stable ?

  3. DISPERSION RELATION (1/2) Neglected in the following Resistive part of the impedance • Stationary distribution • Synchronous phase shift • Potential well distortion Emittance (momentum spread) conservation for protons (leptons)

  4. DISPERSION RELATION (2/2) • Perturbation (around the new fixed point) Linearized Vlasov equation • Dispersion relation Sacherer formula Dispersion integral

  5. STABILITY DIAGRAMS (1/6) • Parabolic • Gaussian Capacitive impedance Below Transition or Inductive impedance Above Transition Reminder

  6. STABILITY DIAGRAMS (2/6) Approximated full spread between centre and edge of the bunch On a flat-top

  7. STABILITY DIAGRAMS (3/6) • Sacherer distribution  Sacherer stability criterion

  8. STABILITY DIAGRAMS (4/6)   

  9. STABILITY DIAGRAMS (5/6) • Synchrotron amplitude distribution • Line density

  10. STABILITY DIAGRAMS (6/6) • What is important for Landau damping (for dipole mode m=1) is

  11. ELLIPTICAL SPECTRUM (1/9)

  12. ELLIPTICAL SPECTRUM (2/9)  S and E curves are very close

  13. ELLIPTICAL SPECTRUM (3/9) Case of the dipole mode m=1 Motions  Instability  Sacherer criterion recovered analytically  Stability criterion Generalization in the presence of frequency spread

  14. ELLIPTICAL SPECTRUM (4/9) Neglecting the synchrotron frequency spread • Dipole mode 

  15. ELLIPTICAL SPECTRUM (5/9) • Quadrupole mode 

  16. ELLIPTICAL SPECTRUM (6/9) Taking into account the synchrotron frequency spread

  17. ELLIPTICAL SPECTRUM (7/9) Reminder : Besnier’s picture (in 1979 for a parabolic bunch) No stability threshold due to the sharp edge of the parabolic distribution  See stability boundary diagram

  18. ELLIPTICAL SPECTRUM (8/9) General stability criterion for mode m  with (Neglecting the synchronous phase shift)

  19. ELLIPTICAL SPECTRUM (9/9)  with

  20. APPLICATION TO THE LHC AT TOP ENERGY (1/2) • The most critical case is the dipole mode m=1  with

  21. APPLICATION TO THE LHC AT TOP ENERGY (2/2) • The previous stability criterion is the same as the one used by Boussard-Brandt-Vos in the paper “Is a longitudinal feedback system required for LHC?” (1999), with • Numerical application with the same parameters as the ones used in the above paper   Same value as the one found by BBV 

  22. ACKNOWLEDGEMENTS • F. Zimmermann, who proposed to look in detail at this mechanism • J. Gareyte, F. Ruggiero, and F. Zimmermann for fruitful discussions

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