Theory of the Synchro-Cyclotron

In the synchro-cyclotron (or frequency-modulated cyclotron) the higher energies available are obtained at the expense of a decrease in the ion current compared with that available from the conventional cyclotron. This decrease results from the fact that during only a small fraction of the frequency-modulation cycle is it possible for ions to be captured into phase stable orbits that do not return to the center during the first phase oscillation. By solving the phase equation, it is possible to obtain a general expression for this fraction, which is defined as the capture efficiency. At a constant dee voltage and varying rate of frequency modulation, the capture efficiency has a maximum at an equilibrium phase angle of 30^ (corresponding to an energy gain per turn equal to half the maximum available). For larger equilibrium phase angles the efficiency decreases as a result of the smaller range of phase stability, while for smaller phase angles it decreases as a result of return of particles to the center. The maximum efficiency is proportional to the square root of the dee voltage or alternatively to the square root of the rate of frequency modulation, and depends on the charge and mass of the ions only through the ratio of charge to mass. Comparisons of the theoretical expectations with available experimental data show satisfactory agreement. Capture efficiencies for present designs of synchro-cyclotrons are of the order of 0. 1 to 2 percent.