Nuclear Physics & Particle Physics
Particle accelerators
Cyclotrons

Cyclotrons

The length limitation can be overcome by making the charged particles follow a circular path. In a cyclotron, charged particles are accelerated across the gap between two 'D' shaped electrodes.

Meanwhile a perpendicular magnetic field moves the particles in a circle. The radius of the circle increases after each successive acceleration, so the path spirals out from the source at the centre to the target on the outside.

We will now derive the unique frequency ƒ at which the accelerating voltage must alternate if the polarity of the D's is to reverse each time the charged particle reaches the gap.

The force on the charge q moving at velocity v perpendicular to magnetic field of strength B is:

F = Bqv

This provides the centripetal force necessary for circular motion:

F = mv²/r
m is the mass of the moving particle.

With v = ω r and ω = 2 π/T = 2 π ƒ , where ω is the particle's angular velocity, T is the orbiting period, and ƒ is the related frequency; we have:


F = Bqv = mv²/r or

Bq = mv/r = m ω =
2 m π ƒ

The frequency is therefore:

ƒ = Bq/ 2πm

Synchrotrons

The relatively small size of cyclotrons makes them useful for producing energetic particles for medical treatments and testing advanced industrial materials.

As the speed of the particles approaches that of light, however, the relativistic mass increase becomes significant and this results in the charges arriving too late to be accelerated across the gap. This puts a limit of about 1 GeV on the energy achievable.

Synchrotrons overcome this problem: the frequency of the alternating voltage increases to match the flight time of the particles.

In a synchrotron, an increasing magnetic field confines a beam of accelerated charged particles to an orbit of fixed radius. Synchrotrons are very large and very expensive machines but they are capable of much higher energies than cyclotrons.

Medical use of a cyclotron

A cyclotron is a machine used to make short-lived radioactive isotopes that can be used for medical imaging or research.

A cyclotron does this by accelerating particles (such as hydrogen atoms) to very high speeds and focusing them on a target substance where a reaction takes place that produces a radioactive element.

Both the accelerated particles and target substance are specifically chosen to produce the desired radioactive element.

Example:

The isotope Fluorine-18 (F-18) is produced in a cyclotron. It is used in the tracer Fluorodeoxyglucose (FDG) which maps glucose metabolism in the body.

Are also produced other radioactive isotopes such as Carbon-11 (C-11), Gallium-68 (Ga-68), and Technetium-99m (Tc-99m) for medical imaging studies.