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# Cyclotron

A cyclotron is a machine designed to accelerate beams of charged particles by using a high frequency alternating voltage across a magnetic field to spiral the beam out and eventually deflect it once the beam's radius equals its container's. At this point the particles' speed is generally very high, approaching the speed of light. The cyclotron was invented by Ernest Lawrence in 1929, who used it in experiments which required particles of speeds of up to 1 MeV. Cyclotrons are used today in the treatment of cancer, as the particles produced ionize tumors and help to stop or slow cancerous growth.

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## Problems solved by the cyclotron

The cyclotron was designed to address the limitations of the linear accelerator, which works by accelerating particles in a straight line through evacuated tubes that contain a series of cylindrical segment electrodes. These electrodes switch from positive to negative voltage repeatedly. At the time it was not possible to get high frequency radio waves at high power, so the stages of acceleration had to be relatively far apart (to accommodate the low frequency) or more stages were required (to compensate for the low power at each stage). To produce higher power required longer accelerators than was economically practical. Later linear accelerators could use devices imparting much more power and at higher frequencies, but without these devices the cyclotron was much more cost effective. The cyclotron was ultimately found to have its own particular limitations. The largest currently operating linear accelerator is the Stanford Linear Accelerator (SLAC), about two miles or 3.2 km long and is far more powerful than the largest cyclotron, due not only to its length but also the use of modern high power and high frequency klystron microwave power tubes. As cyclotrons work by accelerating particles in a circular path they allow much more distance to be covered with a compact accelerator and also allow the effective application of a single and relatively low powered electrical driver. This mechanism does restrict the average power, however.

## Limitations of the cyclotron

While a significant technical achivement at the time, the configuration of the device limits its cost effectiveness at higher power. These limitations were addressed with the invention of the synchrocyclotron (to overcome relativistic effects) and finally the synchrotron, which overcomes the cyclotron's limitations of electromagnet saturation and device size impracticalities imposed by the shape of the vacuum chamber.

## How the cyclotron works

A pair of "Dee" electrodes with loops of coolant pipes on their surface at the Lawrence Hall of Science

The electrodes shown at the right would be contained within the vacuum chamber. The serpentine pipes are for cooling liquid, required since stray particles contact and impart thermal energy to the electrodes. In the cyclotron a magnetic field is applied perpendicular to a disk-shaped vacuum chamber containing a pair of these D shaped semi-circular electrodes. The straight portions of these hollow electrodes are open and face each other. A current of electrons or ions flowing perpendicular to a magnetic field experiences a force that is perpendicular to its direction of motion. (This force is used to practical effect in electric motors.) With charged particles free to move in a vacuum (unlike in a motor where electrons are constrained to the wires), the particles in motion will follow a circular path. If the particles lose energy while circulating they will spiral inward. If the device is capable of applying energy to the particles they will spiral outward. In the cyclotron a high frequency alternating voltage applied across the "D" electrodes causes the particles to accelerate when passing through the gap between the elecrodes. The perpendicular magnetic field forces causes them to travel in a circular path through the D chambers. The particles accelerate only when passing the gap between the two Ds.

## Mathematics of the cyclotron

The centripetal force is provided by the transverse magnetic field B, and the force on a particle travelling in a magnetic field (which causes it to curve) is equal to Bqv. So,

$\frac{mv^2}{r} = Bqv$

(Where m is the mass of the particle, q is its charge, v is its velocity and r is the radius of its path.)

Therefore,

$\frac{v}{r} = \frac{Bq}{m}$

v/r is equal to angular speed, ω, so

$\omega = \frac{Bq}{m}$

And, the frequency

$f = \frac{\omega}{2 \times \pi}$

Therefore,

$f = \frac{Bq}{2 \times \pi \times m}$

This shows that for a particle of constant mass the frequency does not depend on the radius of the particle's orbit. As the beam spirals out its frequency does not decrease and it must continue to accelerate, as it is travelling more distance in the same time. As particles approach the speed of light they acquire additional mass, requiring modifications to the frequency or the magnetic field during the acceleration. This is accomplished in the synchrocyclotron.

## Related technologies

The spiraling of electrons in a cylindrical vacuum chamber within a transverse magnetic field is also employed in the magnetron, a device for producing high frequency radio waves (microwaves).

The Synchrotron moves the particles through a path of constant radius, allowing it to be made as a pipe and so of much larger radius than is practical with the cyclotron and synchrocyclotron.