# Cyclotron

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

A cyclotron is a type of particle accelerator. Cyclotrons accelerate charged particles using a high-frequency, alternating voltage (potential difference). A perpendicular magnetic field causes the particles to go almost in a circle so that they re-encounter the accelerating voltage many times.

Ernest O. Lawrence, of the University of California, Berkeley, is credited with the invention of the cyclotron in 1929. It is less known outside Hungary that Hungarian Sándor Gaál may have described the workings of a cyclotron at about the same time during the spring of 1929 as Lawrence; although almost all reputable international sources give credit to Lawrence for the invention and construction of the first cyclotron. He used it in experiments that required particles with energy of up to 1 MeV.

## How the cyclotron works

Diagram of cyclotron operation from Lawrence's 1934 patent.

The electrodes shown at the right would be in the vacuum chamber, which is flat, in a narrow gap between the two poles of a large magnet.

In the cyclotron, a high-frequency alternating voltage applied across the "D" electrodes (also called "dees") alternately attracts and repels charged particles. The particles accelerate only when passing through the gap between the electrodes. The perpendicular magnetic field (passing vertically through the "D" electrodes) forces the particles to travel in a circular path.

The particles move in a circle, because a current of electrons or ions, flowing perpendicular to a magnetic field, experiences a perpendicular force. The charged particles move freely in a vacuum, so the particles follow a circular path.

## Uses of the Cyclotron

For several decades, cyclotrons were the best source of high-energy beams for nuclear physics experiments; several cyclotrons are still in use for this type of research.

Cyclotrons can be used to treat cancer. Ion beams from cyclotrons can be used, as in proton therapy, to penetrate the body and kill tumors by radiation damage, while minimizing damage to healthy tissue along their path.

Cyclotron beams can be used to bombard other atoms to produce short-lived positron-emitting isotopes suitable for PET imaging.

## Problems solved by the cyclotron

60-inch cyclotron, circa 1939, showing a beam of accelerated ions (likely protons or deuterons) escaping the accelerator and ionizing the surrounding air causing a blue glow. This phenomenon of air ionization is analogous to the one responsible for producing the "blue flash" infamously noted by witnesses of criticality accidents. Though the effect is often mistaken for Cherenkov radiation, this is not the case.

The cyclotron is an improvement over the linear accelerators available when it was invented. A linear accelerator accelerates particles in a straight line, through evacuated tubes. A series of cylindrical electrodes in the tubes switch from positive to negative voltage. In the 1920's, it was not possible to get high frequency radio waves at high power, so the stages of acceleration had to be far apart, to accommodate the low frequency, or more stages were required to compensate for the low power at each stage.

Faster particles required longer accelerators than scientists could afford. Later linear accelerators could use high power Klystrons and other devices imparting much more power at higher frequencies, but before these devices existed, the cyclotron was cheaper.

Cyclotrons accelerate particles in a circular path. Therefore, a compact accelerator can contain much more distance than a linear accelerator, with more opportunities to accelerate the particles.

• Cyclotrons have a single electrical driver, which saves both money and power, since more expense may be allocated to increasing efficiency.
• Cyclotrons produce a continuous stream of particle pulses at the target, so the average power is relatively high.
• The compactness of the device reduces other costs, such as its foundations, radiation shielding, and the enclosing building.

## Limitations of the cyclotron

The magnet portion of a large cyclotron. The gray object is the upper pole piece, routing the magnetic field in two loops through a similar part below. The white canisters held conductive coils to generate the magnetic field. The D electrodes are contained in a vacuum chamber that was inserted in the central field gap.

The spiral path of the cyclotron beam can only "synch up" with klystron-type (constant frequency) voltage sources if the accelerated particles are approximately obeying Newton's Laws of Motion. If the particles become fast enough that relativistic effects become important, the beam gets out of phase with the oscillating electric field, and cannot receive any additional acceleration. The cyclotron is therefore only capable of accelerating particles up to a few percent of the speed of light; higher velocity beams require a synchrocyclotron or a more complex synchrotron or linear accelerator.

## 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

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

Therefore,

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

This shows that for a particle of constant mass, the frequency does not depend upon 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.

The relativistic cyclotron frequency is

$f=f_c\frac{m_0}{m_0+T/c^2}$,

where fc is the classical frequency, given above, of a charged particle with kinetic energy T and rest mass m0 circling in a magnetic field.

The rest mass of an electron is 511 keV, so the frequency correction is 1% for a magnetic vacuum tube with a 5.11 kV direct current accelerating voltage. The proton mass is nearly two thousand times the electron mass, so the 1% correction energy is about 9 MeV, which is sufficient to induce nuclear reactions.

An alternative to the synchrocyclotron is the isochronous cyclotron, which has a magnetic field that increases with radius, rather than with time. The de-focusing effect of this radial field gradient is compensated by ridges on the magnet faces which vary the field azimuthally as well. This allows particles to be accelerated continuously, on every period of the radio frequency, rather than in bursts as in most other accelerator types.

## 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. The larger radius allows the use of numerous magnets, each of which imparts angular momentum and so allowing particles of higher velocity (mass) to be kept within the bounds of the evacuated pipe.