Landau damping dynamic aperture and octupole in LHC

Maximization of the dynamic aperture and Landau damping of the collective instabilities are partly conflicting requirements. On the one hand, the non-linearities of the lattice must be minimized at large oscillation amplitude to guarantee the stability of the single particle motion. On the other hand, a spread of the betatron frequencies is necessary to guarantee the stability of the collective motion of bunches of particles; this requires the introduction of non-linearities effective at small amplitudes. We show in this note that the `natural' spread of betatron tunes due to the field imperfections is inadequate or Landau damping. An octupole scheme is required to provide collective stability at high energy. At low energy it may be used to find the optimum between the correction of the octupolar field imperfections and Landau damping. The solution of the stability problem taking into account the two degrees of freedom of the transverse motion allows a significant saving in octupole strength: 144 octupoles with a magnetic length of 32~cm are needed, leaving some reserve slots for other correctors. The present localization close to the lattice quadrupoles is well adapted for Landau damping, but unfavourable for the correction of the octupolar imperfection of the dipoles. Depending on the multipolar errors in the dipoles, octupolar and skew octupolar spool pieces may have to be envisaged.