# How fast are cyclotrons?

There is a hard limit on an ordinary cyclotron in that they stop working when the particles become relativistic. This limit can be removed with synchrocyclotrons (which change the accelerating frequency as the particles become relativistic), and isochronous cyclotrons (which have a larger magnetic field at a larger radius to account for the relativistic effects).

There is a more important *practical* limit, however. The output energy of a cyclotron scales linearly with the area of the cyclotron, which makes it unreasonably expensive to go to high energies. Because manufacturing larger and larger D's for the cyclotron is more difficult, the cost scales even faster than that, so the cost scales more than linearly with energy. It becomes unreasonably expensive very, very quickly.

The cyclotron depends on the fact that the angular frequency is a constant given by $\omega={qB\over mc}$. However, that equation is in the non-relativistic limit. The correct relativistic equation is $\omega={qB\over mc\gamma}$, so $\omega$ is not a constant when the relativistic parameter gamma increases from its nonrelativistic value of 1. $\gamma$ is related to the energy of the particle be accelerated by the equation $\gamma={T+mc^2\over mc^2}$. This means that the cyclotron will stop working when the kinetic energy T becomes too large. That is, the cyclotron requires that ${T\over mc^2}\ll1$.