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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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  • $\begingroup$ What is an "average" cyclotron? Ordinary cyclotrons do not do relativistic particles. But then there are synchrocyclotrons. $\endgroup$ – Pieter May 6 '18 at 21:25
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    $\begingroup$ "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. ;-) $\endgroup$ – uhoh May 7 '18 at 10:33
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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.

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  • $\begingroup$ re "...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)? $\endgroup$ – uhoh May 7 '18 at 7:10
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    $\begingroup$ @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. $\endgroup$ – Chris May 7 '18 at 7:31
  • $\begingroup$ Regarding the practical limit: Wikipedia lists TRIUMF as the largest cyclotron with 500 MeV $\endgroup$ – Polygnome May 7 '18 at 8:01
  • $\begingroup$ @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. $\endgroup$ – Jon Custer May 7 '18 at 13:10
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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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In the spirit of a vague question, a typical cyclotron will accelerate a particle from injection energy to extraction energy in 100 to 1000 turns. Depends of the details, of course, and that's just a ballpark figure. To get from turns to seconds you need the cyclotron frequency, which is also the RF frequency up to an integer harmonic number, but we're usually talking MegaHertz. The limitations are that the particles can't be accelerated to relativistic energies, that you need a large and/or strong (but either way that means expensive) magnet, and you need to focus the particles to stop them drifting transversely and hitting the walls.

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Beyond relativity (which is overcome by making the cyclotron isochronous) and cost issues, there is a fundamental limitation which comes from the extraction area. There, the beam at the energy that you desire and the beam at the previous turn, need to be spatially separated in order to be able to extract only the former. There are some tricks that can be played by introducing orbit bumps and adjusting the tune of the machine, but ultimately the lack of spatial separation will totally impede the extraction.

This is why superconducting cyclotrons will never EVER get as large as their energy-record-holding normal-conducting counterparts, which would in principle not be so crazy in the LHC and ITER era.

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