Long-term stability studies for CERN-LHC

In the modern hadron colliders, like LHC, SSC and RHIC, the stability of the single-particle motion is basically determined by the field-shape imperfections of the superconducting dipoles and quadrupoles, especially during the injection flat bottom, when the effect of the persistent currents is maximum and the transverse size of the beam is large. The non-linear fields are at the origin of two effects: the betatron tunes change with the amplitude and the momentum of the circulating particles, and, for certain combinations of the horizontal, vertical, and synchrotron tunes, non-linear resonances are excited. These phenomena have a destabilizing influence on the particle motion, over a time-scale extending up to several million turns. Some precautions can make the motion of the particles less sensitive to the non-linear components of the guiding fields. Correcting multipoles can be foreseen in the regular cells, to reduce the non-linear tune-shift caused by the systematic components of the field errors. The variations of the orbit functions can be limited along the insertions. The closed orbit and the linear coupling can be corrected sufficiently well. Finally the ripple of the power supplies can be reduced as much as possible. Most of these concepts have been embedded in the design of the LHC and their beneficial effects on the dynamic aperture have been extensively evaluated by computer simulations