Cyclotrons

For other uses, see Cyclotron (disambiguation).

A cyclotron is a type of particle accelerator in which charged particles accelerate outwards from the center along a spiral path. The particles are held to a spiral trajectory by a static magnetic field and accelerated by a rapidly varying (radio frequency) electric field.

History

The cyclotron was invented and patented by Ernest Lawrence of the University of California, Berkeley, where it was first operated in 1932. A graduate student, M. Stanley Livingston, did much of the work of translating the idea into working hardware. Lawrence read an article about the concept of a drift tube linac by Rolf Widerøe, who had also been working along similar lines with the betatron concept. The first European cyclotron was constructed in Leningrad in the physics department of the Radium Institute, headed by Vitaly Khlopin (ru). This instrument was first proposed in 1932 by George Gamow and Lev Mysovskii (ru) and was installed and became operative by 1937.

Principle of operation

Cyclotrons accelerate charged particle beams using a high frequency alternating voltage which is applied between two "D"-shaped electrodes (also called "dees"). An additional static magnetic field $B$ is applied in perpendicular direction to the electrode plane, enabling particles to re-encounter the accelerating voltage many times at the same phase. To achieve this, the voltage frequency must match the particle's cyclotron resonance frequency

$f = \frac\left\{q B\right\}\left\{2\pi m\right\}$,

with the relativistic mass m and its charge q. This frequency is given by equality of centripetal force and magnetic Lorentz force. The particles, injected near the centre of the magnetic field, increase their kinetic energy only when recirculating through the gap between the electrodes; thus they travel outwards along a spiral path. Their radius will increase until the particles hit a target at the perimeter of the vacuum chamber, or leave the cyclotron using a beam tube, enabling their use e.g. for particle therapy. Various materials may be used for a target, and the collisions will create secondary particles which may be guided outside of the cyclotron and into instruments for analysis.

Relativistic considerations

In the nonrelativistic approximation, the frequency does not depend upon the radius of the particle's orbit, since the particle's mass is constant. As the beam spirals out, its frequency does not decrease, and it must continue to accelerate, as it is travelling a greater distance in the same time period. In contrast to this approximation, as particles approach the speed of light, their relativistic mass increases, requiring either modifications to the frequency, leading to the synchrocyclotron, or modifications to the magnetic field during the acceleration, which leads to the isochronous cyclotron. The relativistic mass can be rewritten as

$m = \frac\left\{m_0\right\}\left\{\sqrt\left\{1-\left\left(\frac\left\{v\right\}\left\{c\right\}\right\right)^2\right\}\right\} = \frac\left\{m_0\right\}\left\{\sqrt\left\{1-\beta^2\right\}\right\} = \gamma \left\{m_0\right\}$,

where

$m_0$ is the particle rest mass,
$\beta = \frac\left\{v\right\}\left\{c\right\}$ is the relative velocity, and
$\gamma=\frac\left\{1\right\}\left\{\sqrt\left\{1-\beta^2\right\}\right\}=\frac\left\{1\right\}\left\{\sqrt\left\{1-\left\left(\frac\left\{v\right\}\left\{c\right\}\right\right)^2\right\}\right\}$ is the Lorentz factor.

The relativistic cyclotron frequency and angular frequency can be rewritten as

$f = \frac\left\{q B\right\}\left\{2\pi \gamma m_0\right\} = \frac\left\{f_0\right\}\left\{\gamma\right\} = \left\{f_0\right\}\left\{\sqrt\left\{1-\beta^2\right\}\right\} = \left\{f_0\right\}\left\{\sqrt\left\{1-\left\left(\frac\left\{v\right\}\left\{c\right\}\right\right)^2\right\}\right\}$, and
$\omega = \left\{2\pi f\right\} = \frac\left\{q B\right\}\left\{\gamma m_0\right\} = \frac\left\{\omega_0\right\}\left\{\gamma\right\} = \left\{\omega_0\right\}\left\{\sqrt\left\{1-\beta^2\right\}\right\} = \left\{\omega_0\right\}\left\{\sqrt\left\{1-\left\left(\frac\left\{v\right\}\left\{c\right\}\right\right)^2\right\}\right\}$,

where

$f_0$ would be the cyclotron frequency in classical approximation,
$\omega_0$ would be the cyclotron angular frequency in classical approximation.

The gyroradius for a particle moving in a static magnetic field is then given by

$r = \frac\left\{v\right\}\left\{\omega\right\} = \frac\left\{\beta c\right\}\left\{\omega\right\} = \frac\left\{\gamma \beta m_0 c\right\}\left\{q B\right\}$,

because

$\omega r = v = \beta c$

where v would be the (linear) velocity.

Synchrocyclotron

Main article: Synchrocyclotron

A synchrocyclotron is a cyclotron in which the frequency of the driving RF electric field is varied to compensate for relativistic effects as the particles' velocity begins to approach the speed of light. This is in contrast to the classical cyclotron, where the frequency was held constant, thus leading to the synchrocyclotron operation frequency being

$f = \frac\left\{f_0\right\}\left\{\gamma\right\} = \left\{f_0\right\}\left\{\sqrt\left\{1-\beta^2\right\}\right\}$,

where $f_0$ is the classical cyclotron frequency and $\beta = \frac\left\{v\right\}\left\{c\right\}$ again is the relative velocity of the particle beam. The rest mass of an electron is 511 keV/c2, so the frequency correction is 1% for a magnetic vacuum tube with a 5.11 keV/c2 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.

Isochronous cyclotron

An alternative to the synchrocyclotron is the isochronous cyclotron, which has a magnetic field that increases with radius, rather than with time. Isochronous cyclotrons are capable of producing much greater beam current than synchrocyclotrons, but require azimuthal variations in the field strength to provide a strong focusing effect and keep the particles captured in their spiral trajectory. For this reason, an isochronous cyclotron is also called an "AVF (azimuthal varying field) cyclotron". This solution for focusing the particle beam was proposed by L. H. Thomas in 1938. Recalling the relativistic gyroradius $r = \frac\left\{\gamma m_0 v\right\}\left\{q B\right\}$ and the relativistic cyclotron frequency $f = \frac\left\{f_0\right\}\left\{\gamma\right\}$, one can choose $B$ to be proportional to the Lorentz factor, $B = \gamma B_0$. This results in the relation $r = \frac\left\{m_0 v\right\}\left\{q B_0\right\}$ which again only depends on the velocity $v$, like in the non-relativistic case. Also, the cyclotron frequency is constant in this case.

The transverse 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 (RF), rather than in bursts as in most other accelerator types. This principle that alternating field gradients have a net focusing effect is called strong focusing. It was obscurely known theoretically long before it was put into practice. Examples of isochronous cyclotrons abound; in fact almost all modern cyclotrons use azimuthally-varying fields. The TRIUMF cyclotron mentioned below is the largest with an outer orbit radius of 7.9 metres, extracting protons at up to 510 MeV, which is 3/4 of the speed of light. The PSI cyclotron reaches higher energy but is smaller because of using a higher magnetic field.

Usage

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. The results enable the calculation of various properties, such as the mean spacing between atoms and the creation of various collision products. Subsequent chemical and particle analysis of the target material may give insight into nuclear transmutation of the elements used in the target.

Cyclotrons can be used in particle therapy 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. More recently cyclotrons currently installed at hospitals for particle therapy have been retrofitted to enable them to produce technetium-99. Technetium-99 is a diagnostic isotope in short supply due to difficulties at Canada's Chalk River facility.

The cyclotron was an improvement over the linear accelerators (linacs) that were available when it was invented, being more cost- and space-effective due to the iterated interaction of the particles with the accelerating field. In the 1920s, it was not possible to generate the high power, high-frequency radio waves which are used in modern linacs (generated by klystrons). As such, impractically long linac structures were required for higher-energy particles. The compactness of the cyclotron reduces other costs as well, such as foundations, radiation shielding, and the enclosing building. Cyclotrons have a single electrical driver, which saves both money and power. Furthermore, cyclotrons are able to produce a continuous stream of particles at the target, so the average power passed from a particle beam into a target is relatively high.

The spiral path of the cyclotron beam can only "sync 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 becomes out of phase with the oscillating electric field, and cannot receive any additional acceleration. The classical cyclotron is therefore only capable of accelerating particles up to a few percent of the speed of light. To accommodate increased mass the magnetic field may be modified by appropriately shaping the pole pieces as in the isochronous cyclotrons, operating in a pulsed mode and changing the frequency applied to the dees as in the synchrocyclotrons, either of which is limited by the diminishing cost effectiveness of making larger machines. Cost limitations have been overcome by employing the more complex synchrotron or modern, klystron-driven linear accelerators, both of which have the advantage of scalability, offering more power within an improved cost structure as the machines are made larger.

Notable examples

The world's largest cyclotron is at the RIKEN laboratory in Japan. Called the SRC, for Superconducting Ring Cyclotron, it has 6 separated superconducting sectors, and is 19 m in diameter and 8 m high. Built to accelerate heavy ions, its maximum magnetic field is 3.8 tesla, yielding a bending ability of 8 tesla-metres. The total weight of the cyclotron is 8,300 tonnes. It has accelerated uranium ions to 345 MeV per atomic mass unit.

TRIUMF, Canada's national laboratory for nuclear and particle physics, houses one of the world's largest cyclotrons. The 18 m diameter, 4,000 tonne main magnet produces a field of 0.46 T while a 23 MHz 94 kV electric field is used to accelerate the 300 μA beam. Its large size is partly a result of using negative hydrogen ions rather than protons. The advantage is that extraction is simpler; multi-energy, multi-beams can be extracted by inserting thin carbon stripping foils at appropriate radii. The disadvantage is that the magnetic field is limited: a magnetic field larger than about 0.5 tesla can prematurely strip the loosely-bound second electron. TRIUMF is run by a consortium of eighteen Canadian universities and is located at the University of British Columbia, Vancouver, Canada.

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 allows particles of higher velocity (mass) to be kept within the bounds of the evacuated pipe. The magnetic field strength of each of the bending magnets is increased as the particles gain energy in order to keep the bending angle constant.