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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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