The Paths of Ions in the Cyclotron I. Orbits in the Magnetic Field

Bethe and Rose maintain in a recent letter and paper that a maximum energy for the beam from a cyclotron is fixed by the incompatibility of the conditions for resonance and focusing when the relativity increase of mass with velocity is taken into account. It is shown below that, while this result holds for a radially symmetrical magnetic field, it is not necessarily true in general; and that for a field varying with polar angle there is an additional focusing effect. If the relative variation of the field with polar angle is of the order of the ratio of the velocity of the ion to the velocity of light, this focusing effect will compensate the defocusing effect of Bethe and Rose. It is shown further that if this variation has period π2, a family of stable periodic orbits exists which are nearly concentric circles. The second order effects due to the simultaneous action of the variations with polar angle of the magnetic field and the accelerating electric field will be considered in a second paper.