A New Code for Simulation of Space Charge Effects in Isochronous Cyclotrons

The paper describes a new beam dynamics code for simulation of space charge effects in isochronous cyclotrons. The program assumes that the beam consists of many point macro-particles. The code uses a set of simplified equations of motion. It adopts continuous and slow acceleration approximation that allows one to separate betatron motion from slower energy-phase motion. Thus, the particle radial displacement r can be described as r=R+x where R is the slowly changing part due to energy-phase motion and x is the betatron part. To further simplify the problem we use the smooth focusing approximation: Azimuthally dependent focusing is substituted by its average value calculated over one turn. Thus, focusing depends on only particle s energy and can be expressed via betatron frequencies νr , νz. Besides, R is substituted by its average value. Such parameters of a trajectory as R, νr, νz, and particle rotation frequency are found from a precalculated table of the equilibrium orbit parameters. The table includes equilibrium orbit parameters calculated for a coasting beam with zero current for different energies. The table is prepared using measured or calculated cyclotron median plane magnetic field. The equations of motion are (in the CGS unit system):