The cyclotron uses electric and magnetic fields. The whole accelerator is in a uniform magnetic field. Charged particles moving in the field feel a force acting at 90^o to their direction of motion, so they move in circles – the Lorentz force due to the field provides the necessary centripetal force for circular motion:

giving

so the particles move in circles with radii proportional to their speed.

The time taken for one orbit is

and so the orbit frequency is

Similar particles (i.e. with equal charges and masses) will all orbit at the same rate, regardless of their speed.

The particles orbit inside two metal semicircles called Dees. These are connected to opposite terminals of an a.c. supply set to the cyclotron frequency calculated for the particles and magnetic field being used. The particles are accelerated in bunches by the electric field between the plates and, as all particles have the same orbit frequency, they will all be accelerated by the field. As the particles accelerate they move in larger and larger circles until reaching a deflector at the edge of the machine which directs them to the target.

Their maximum speed as they reach the outside is and therefore their maximum energy is

Lawrence and Livingston built the first cyclotron in 1932. It was about 30 cm across, in a magnetic field of about 0.5 T and accelerated protons to roughly 1.2 MeV.

The frequency of the a.c. supply depends on the specific charge of the particles (e/m) and the strength of the magnetic field. To obtain as large a KE_max as possible, the machine should have as large a radius as possible and as large a magnetic field as possible. However there are problems producing strong magnetic fields which have a constant value over the large areas needed. As the magnetic field increases so does the cyclotron frequency, already in the MHz for the first machines.

The real limiting factor is the basic design – all particles must orbit at the same frequency, whatever their speed. As particles approach the speed of light, however, they behave as if their mass is increasing. Accelerating them becomes more difficult and they start to lag behind the oscillating electric field. As cyclotrons approached 20 MeV they began to reach their limits and a new design had to be produced.

A particle becomes relativistic once its kinetic energy is comparable to its rest energy (E_rest = mc²). As the rest energy of the electron is only 500 keV they cannot be accelerated to a useful energy in a cyclotron.

Introduction        Direct voltage and cascade machines           Cyclotrons

Betatrons                    Linear accelerators and the synchrocyclotron                Synchrotrons       

    Fixed target verses collider machines        Lepton verses Hadron machines        The Future?