1 INTRODUCTION
In high intensity rings, the size of the beam is enhanced to avoid excessive space-charge forces. The beam thus occupies the highly nonlinear region of the magnetic field near the bore and most of the RF bucket. Large fringe fields are inherent to short and wide magnets and decrease further the beam stability [1]. Rapid acceleration produces non-adiabatic trapping on the longitudinal plane. All these effects pump protons from the core of the beam into the tails in all three dimensions. On the other hand, the relative aperture is dramatically small. While in a high energy collider the aperture of the accelerator is around (where is rms beam size), in a high intensity ring the aperture is typically less than .