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
Elias Metral, ABP Forum, 26/02/2003

2. Incoherent synchrotron frequency shift
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 ?
Elias Metral, ABP Forum, 26/02/2003

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)
Elias Metral, ABP Forum, 26/02/2003

4. DISPERSION RELATION (2/2)
Perturbation (around the new fixed point)  Linearized Vlasov equation
Dispersion relation
Sacherer formula
Dispersion integral
Elias Metral, ABP Forum, 26/02/2003

5. STABILITY DIAGRAMS (1/6)
Parabolic
Gaussian
Capacitive impedance Below Transition or Inductive impedance Above Transition
Reminder
Elias Metral, ABP Forum, 26/02/2003

6. STABILITY DIAGRAMS (2/6)
Approximated full spread between centre and edge of the bunch
On a flat-top
Elias Metral, ABP Forum, 26/02/2003

7. STABILITY DIAGRAMS (3/6)
Sacherer distribution 
Sacherer stability criterion
Elias Metral, ABP Forum, 26/02/2003

8. STABILITY DIAGRAMS (4/6)  
Elias Metral, ABP Forum, 26/02/2003

9. STABILITY DIAGRAMS (5/6)
Synchrotron amplitude distribution
Line density
Elias Metral, ABP Forum, 26/02/2003

10. STABILITY DIAGRAMS (6/6)
What is important for Landau damping (for dipole mode m = 1) is
Elias Metral, ABP Forum, 26/02/2003

11. ELLIPTICAL SPECTRUM (1/9)
Elias Metral, ABP Forum, 26/02/2003

12. ELLIPTICAL SPECTRUM (2/9)
 S and E curves are very close
Elias Metral, ABP Forum, 26/02/2003

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
Elias Metral, ABP Forum, 26/02/2003

14. ELLIPTICAL SPECTRUM (4/9)
Neglecting the synchrotron frequency spread
Dipole mode 
Elias Metral, ABP Forum, 26/02/2003

15. ELLIPTICAL SPECTRUM (5/9)
Quadrupole mode 
Elias Metral, ABP Forum, 26/02/2003

16. ELLIPTICAL SPECTRUM (6/9)
Taking into account the synchrotron frequency spread
Elias Metral, ABP Forum, 26/02/2003

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
Elias Metral, ABP Forum, 26/02/2003

18. ELLIPTICAL SPECTRUM (8/9)
General stability criterion for mode m 
with
(Neglecting the synchronous phase shift)
Elias Metral, ABP Forum, 26/02/2003

19. ELLIPTICAL SPECTRUM (9/9)
with
Elias Metral, ABP Forum, 26/02/2003

20. APPLICATION TO THE LHC AT TOP ENERGY (1/2)
The most critical case is the dipole mode m = 1 
with
Elias Metral, ABP Forum, 26/02/2003

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 
Elias Metral, ABP Forum, 26/02/2003

22. F. Zimmermann, who proposed to look in detail at this mechanism
ACKNOWLEDGEMENTS
F. Zimmermann, who proposed to look in detail at this mechanism
J. Gareyte, F. Ruggiero, and F. Zimmermann for fruitful discussions
Elias Metral, ABP Forum, 26/02/2003