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Stability of the longitudinal coherent modes

Stability of the longitudinal coherent modes. General stability criterion Application to the LHC at top energy. General stability criterion (1/6). Stationary distribution. Synchronous phase shift. Potential well distortion. Perturbation (around the new fixed point). Dispersion relation.

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Stability of the longitudinal coherent modes

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  1. Stability of the longitudinalcoherent modes • General stability criterion • Application to the LHC at top energy

  2. General stability criterion (1/6) • Stationary distribution • Synchronous phase shift • Potential well distortion • Perturbation (around the new fixed point) • Dispersion relation Sacherer formula Dispersion integral

  3. General stability criterion (2/6) • Distribution of the normalised synchrotron amplitudes • Stability diagram

  4. General stability criterion (3/6) • The stability boundary can be approximated by semicircles (following the example of Keil-Schnell for coasting beams) Sacherer stability criterion is the approximated full spread between centre and edge of the bunch on a flat-top

  5. General stability criterion (4/6) At low intensity and neglecting synchrotron frequency spread • Dipole mode is the incoherent small-amplitude frequency shift • Quadrupole mode

  6. General stability criterion (5/6) • Stability of the dipole mode Stability criterion discussed during the last LCE meeting  with

  7. General stability criterion (6/6)  with

  8. Application to the LHC at top energy • 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 (0.28  for 00 and 11, …) Value found by BBV with

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