Cyclotron

Image:LawrenceCyclotronDees.jpg A cyclotron accelerates charged particles with a high-frequency, alternating voltage (potential difference). A perpendicular magnetic field causes the particles to go almost in a circle. The beam spirals out to the edge of the container, as the particles' speeds increase. At this point, the particles' speed approaches the speed of light.

The cyclotron was invented by Ernest Lawrence of the University of California, in 1929. He used it in experiments that required particles with energy of up to 1 MeV. Cyclotrons are used today to treat cancer. The bright adjustable-frequency x-rays produced by a cyclotron's radiation can be adjusted to penetrate limited distances into the human body, to kill tumor cells.

How the cyclotron works

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 (which are also called "dees") alternately attracts and repels charged particles. So, the particles accelerate only when passing through the gap between the electrodes. The perpendicular magnetic field (going "down" through the top of the "D" electrodes) forces the particles to travel in a circular path through the D-shaped chambers in the electrodes.

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.

If the particles slow down (lose energy) they will spiral inward. If the device applies energy to the particles, they will speed up, and spiral outward.

The serpentine pipes in the electrodes carry cooling liquid to remove the heat that is caused when stray particles hit the electrodes.

Problems solved by the cyclotron

Image:Cyclotron with glowing beam.jpg The cyclotron is an improvement of the linear accelerator. 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

Image:LawrenceCyclotronMagnet.jpg The cyclotron has its own limitations. As the beam speed increases, cyclotron radiation is emitted from the side of the beam, because the magnet is turning and slowing, ("braking") the beam. Small cyclotrons with fast beams can waste all of their energy generating radiation at higher beam speeds.

In research cyclotrons that accelerated particle beams, the electrical driver was usually sized and powered so that most of its energy was dissipated by generating radiation, with relatively few, very high energy particles. As a result, the cyclotron is usually shielded, so that the operators are not harmed by the x-rays it emits.

Most modern cyclotrons are constructed especially to produce bremsstrahlung radiation. Cyclotrons produce spectrally-pure, very-bright far-ultraviolet(λ=less than 400nm), and soft, low-frequency x-rays, that are difficult to produce by other methods.

While a significant technical achievement at the time, cyclotrons are too expensive at higher powers. Their limitations caused the invention of the synchrocyclotron (to overcome relativistic effects), and finally the synchrotron, which overcomes the cyclotron's limitations: that the electromagnet saturates (becoming unable to produce additional magnetic field with additional coil current), and larger cyclotrons are much too large because of the shape of their vacuum chambers.

Large linear accelerators do not have bremsstrahlung radiation, because the beam does not change direction. The largest modern linear accelerator is the Stanford Linear Accelerator (SLAC), about two miles (3.2 km) long. It is far more powerful than the largest cyclotron. This is due not only to its length, and straightness, but also to the use of modern high-power and high-frequency klystron microwave power tubes.

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 $f_c$ is the classical frequency, given above, of a charged particle with kinetic energy $T$ and rest mass $m_0$ 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.