Editing Cyclotron (section)

==== Isochronous cyclotron ====
[[File:Lorentz factor.svg|thumb|250px|In isochronous cyclotrons, the magnetic field strength {{mvar|B}} as a function of the radius {{mvar|r}} has the same shape as the Lorentz factor {{mvar|γ}} as a function of the speed {{mvar|v}}.]]

If instead the magnetic field is varied with radius while the frequency of the accelerating field is held constant, this leads to the ''isochronous cyclotron''.{{r|wilson}}

<math display="block">B(r) = \frac{m_0}{q \sqrt{\left(\frac{1}{2\pi f}\right)^2 - \left(\frac{r}{c}\right)^2}}</math>

Keeping the frequency constant allows isochronous cyclotrons to operate in a continuous mode, which makes them capable of producing much greater beam current than synchrocyclotrons. On the other hand, as precise matching of the orbital frequency to the accelerating field frequency is the responsibility of the magnetic field variation with radius, the variation must be precisely tuned.