An analytical envelope model for space charge in cyclotrons

Abstract

Abstract The statistical σ-matrix approach is used to derive general envelope equations for ellipsoidal bunches with space charge in isochronous and non-isochronous fixed-field accelerators (FFAs) These accelerators couple the radial and longitudinal phase spaces due to momentum dispersion induced by the space charge effect. This generates a rotation of the bunch around the local vertical axis which is known in the field as the vortex effect. A special invariant of the vortex is found which represents its total angular momentum with respect to its center. For the isochronous cyclotron, a special solution of the envelope equations is found which describes a circular bunch with equal radial and longitudinal RMS sizes. The existence of this solution is a direct consequence of the circular symmetry of the single particle Hamiltonian in a co-moving coordinate frame. The stationary solution of the circular bunch fixes the relation between bunch sizes and accelerated current (or the total charge per bunch) for given emittances. The general envelope equations may easily be implemented in existing numerical envelope codes.