Cyclotron Spiral Inflector

The most common device for injecting into a compact cyclotron is an electrostatic device known as the spiral inflector. There is a known analytic solution for the trajectory, and therefore the electrode shape, in the idealized case of a uniform magnetic field. This is re-derived in this note and the conditions for perfect centring are found. It is moreover found that the ‘Muller’, or hyperboloid inflector is actually a special case of the spiral inflector.