Space Charge Effects and Current Limitations in Cyclotrons

The effects of the electric fields generated by the ion beam in a cyclotron are investigated theoretically on the assumption that the beam consists of well defined bursts of ions moving on separated orbits. The theoretical analysis is carried out in two steps by first calculating the field acting on an ion at the surface of an individual burst due to the charge contained in the burst itself and then adding the contributions of charges on neighboring orbits. The electric field is found to be a function of the burst´s radial position, inversely proportional to the velocity (or radius) on the first turns and then rapidly leveling off at intermediate radii. The influence of various parameters (dimensions of the bursts, energy gain per turn, etc.), the changes in the betatron frequencies and current limitations due to the space charge forces are discussed and illustrated in a numerical example.