# Cyclotron | relativistic considerations

## Relativistic considerations

A French cyclotron, produced in Zurich, Switzerland in 1937. The vacuum chamber containing the dees (at left) has been removed from the magnet (red, at right)

In the non-relativistic approximation, the cyclotron frequency does not depend upon the particle's speed or the radius of the particle's orbit. As the beam spirals out, the rotation frequency stays constant, and the beam continues to accelerate as it travels a greater distance in the same time period.

In contrast to this approximation, as particles approach the speed of light, the cyclotron frequency decreases proportionally to the particle's Lorentz factor. A rigorous proof of this fact (starting from Newton's second law) is given here: Relativistic_mechanics#Force. Acceleration of relativistic particles therefore requires either modification to the frequency during the acceleration, leading to the synchrocyclotron, or modification to the magnetic field during the acceleration, which leads to the isochronous cyclotron. The relativistic mass can be rewritten as

${\displaystyle m={\frac {m_{0}}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}={\frac {m_{0}}{\sqrt {1-\beta ^{2}}}}=\gamma {m_{0}}}$,

where

${\displaystyle m_{0}}$ is the particle rest mass,
${\displaystyle \beta ={\frac {v}{c}}}$ is the relative velocity, and
${\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}={\frac {1}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}}$ is the Lorentz factor.

The relativistic cyclotron frequency and angular frequency can be rewritten as

${\displaystyle f={\frac {qB}{2\pi \gamma m_{0}}}={\frac {f_{0}}{\gamma }}={f_{0}}{\sqrt {1-\beta ^{2}}}={f_{0}}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}$, and
${\displaystyle \omega ={2\pi f}={\frac {qB}{\gamma m_{0}}}={\frac {\omega _{0}}{\gamma }}={\omega _{0}}{\sqrt {1-\beta ^{2}}}={\omega _{0}}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}$,

where

${\displaystyle f_{0}}$ would be the cyclotron frequency in classical approximation,
${\displaystyle \omega _{0}}$ would be the cyclotron angular frequency in classical approximation.

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

${\displaystyle r={\frac {v}{\omega }}={\frac {\beta c}{\omega }}={\frac {\gamma \beta m_{0}c}{qB}}}$,

because

${\displaystyle \omega r=v=\beta c}$

where v would be the (linear) velocity.

### 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

${\displaystyle f={\frac {f_{0}}{\gamma }}={f_{0}}{\sqrt {1-\beta ^{2}}}}$,

where ${\displaystyle f_{0}}$ is the classical cyclotron frequency and ${\displaystyle \beta ={\frac {v}{c}}}$ 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 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.

### 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".[19] This solution for focusing the particle beam was proposed by L. H. Thomas in 1938.[19] Recalling the relativistic gyroradius ${\displaystyle r={\frac {\gamma m_{0}v}{qB}}}$ and the relativistic cyclotron frequency ${\displaystyle f={\frac {f_{0}}{\gamma }}}$, one can choose ${\displaystyle B}$ to be proportional to the Lorentz factor, ${\displaystyle B=\gamma B_{0}}$. This results in the relation ${\displaystyle r={\frac {m_{0}v}{qB_{0}}}}$ which again only depends on the velocity ${\displaystyle v}$, 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.[20] 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.