Space-Charge Forces in Strong-Focusing Synchrotrons

Abstract

A new principle, strong focusing, has recently been put forward as the basis for the design of synchrotrons, particularly proton synchrotrons in the energy range above about 10 Bev. In this new proposal the field gradient index, $n$, is alternatively positive and negative and has a numerical value of some thousands as opposed to conventional, weak-focusing synchrotrons and betatrons, where $0lnl1$. The object of this note is to examine the effect of space-charge forces in strong-focusing synchrotrons on the stability of the radial and axial motion of the particles being accelerated.It is shown that, by virtue of the decrease of these forces as the particle velocity approaches that of light, the average field index is gradually altered throughout the cycle of acceleration. Such modulation, if sufficiently large, can make it impossible to provide radial and axial stability for the whole of this cycle. If the number of sectors, $N$, is much greater than 10, however, much smaller amounts of modulation may create conditions favorable for forced radial or axial resonances in the presence of inevitable magnet alignment errors, and it is this effect which places stringent limits on the strength of space charge forces permitted in a strong-focusing accelerator. The limitations discussed are far more serious in the case of proton beams than in the case of electron beams.