How fast are cyclotrons?
Solution 1
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.
Solution 2
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 ${T\over mc^2}\ll1$.
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portaloo
Updated on July 27, 2020Comments
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portaloo over 3 years
I know this question is kind of vague, but I just want have a handle on how fast a average cyclotron can accelerate particles, and what kind of limits there are...
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Admin over 5 yearsWhat is an "average" cyclotron? Ordinary cyclotrons do not do relativistic particles. But then there are synchrocyclotrons.
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uhoh over 5 years"How fast are cyclotrons?" All cyclotrons I know of are moving at about 30 km/sec around the Sun. That may change some day, but for now it's a good estimate. ;-)
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uhoh over 5 yearsre "...manufacturing larger and larger D's..." Aren't the D's just conducting RF electrodes? Isn't it the size of the magnet that is the major cost (steel and either electrical power + copper or superconducting wire + refrigerator + helium)?
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Chris over 5 years@uhoh "Aren't the D's just conducting RF electrodes?" Almost. They're conducting RF electrodes under vacuum. That costs a pretty penny. Though I was lumping the cost of the magnets into the "D's" as well. The main point is just the cost goes up more than 2x if you double the area, as opposed to a synchrotron, which can be made piecemeal and so you can actually enjoy some economies of scale and it's possible the cost could scale less than linearly.
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Polygnome over 5 yearsRegarding the practical limit: Wikipedia lists TRIUMF as the largest cyclotron with 500 MeV
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Jon Custer over 5 years@Polygnome - going by cyclotron K-values, TRIUMF is not unique in being a K500 machine (Texas A&M has one for instance). There are other K500 machines in the world, and I know larger ones (K1200) were built (Michigan State) - not sure about the operational status of various ones these days.