Accelerators with similar orbits

Abstract

We obtain the conditions that must be satisfied by a magnetic system in order that the frequencies of radial and vertical betatron oscillations be independent of the particle momenta (in this case the orbits are called dynamically similar). In such systems mere should in principle be no excitation of betatron oscillations associated with synchrotron oscillations and other phenomena. A magnetic field$H_0 \left( {\theta } \right)\left( {{{r_0 } \mathord{\left/ {\vphantom {{r_0 } r}} \right. \kern-\nulldelimiterspace} r}} \right)^{\Pi _0 }$ with n₀ = const produces both geometric and dynamic similarity of the orbit. In weak-focusing accelerators with segments (race tracks) and in strong-focusing proton synchrotrons, the orbits are not dynamically similar. In order to obtain this kind of similarity in the first case, in addition to n₀ = const it is necessary that the magnet sectors have a common center. Different types of annular synchrocyclotrons are considered. In the first type the centers of neighboring magnet sectors are located on different sides of the doughnut and in the second type at the same point (at the center of the accelerator). In the second type the orbits are dynamically similar, unlike those of the first. It is shown mat it is possible to design an annular synchrocyclotron in which the particles can move with stability simultaneously in both directions within the doughnut.