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In John Gribbin's Ten Tantalizing Truths, the author discusses the cyclotron:

for a particular mass as the particles spiral outward, the rotation frequency stays constant, because the particles are traveling faster over a longer distance, round a bigger circle. So the frequency of the kick provided by the electric field stays the same and the beam continues to accelerate. But this sets a limit on how fast the particles can go. Once they are going at a sizeable fraction of the speed of light, the extra energy they receive from each kick by the electric field doesn’t simply go into making them move faster; it makes them more massive. But the cyclotron frequency depends on the mass, so they get out of step with the varying electric field. This is unfortunate because the extra mass is a measure of how much more energy they are carrying, so even if the speed doesn’t increase by much, they pack a more powerful punch when they hit the target, which is what the experimenters want.

How can it be "unfortunate" while this is what the experiments want?

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    $\begingroup$ It is "unfortunate" because it means that the cyclotron cannot accelerate them further. For more energy you need to be able to adjust the frequency (synchrotron) and this is complicated. $\endgroup$
    – mike stone
    Jul 17, 2023 at 11:37
  • $\begingroup$ @mikestone But why is this needed while they already pack a desired powerful punch? $\endgroup$ Jul 17, 2023 at 11:41
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    $\begingroup$ @AhmedSamir The higher the energy of the particles, the more we can learn about particle physics from their collisions. There isn't any point where physicists go "ehh... we've achieved our desired powerful punch, that's enough energy; we can stop there" $\endgroup$
    – AXensen
    Jul 17, 2023 at 12:33
  • $\begingroup$ A strange statement as the "energy" is not lost. It is still there and available for use to smashing up particles. $\endgroup$
    – Farcher
    Jul 17, 2023 at 14:22
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    $\begingroup$ I think this is just a simple misinterpretation of the english language here. The unfortunate thing is that they get out of step (previous sentence), what the experiment wants is to "pack a more powerful punch". What is confusing in my opinion is that mass or speed is the same here, it is just that you can't increase any of those two beyond a certain threshold due to technical limitations of the setup. $\endgroup$ Jul 18, 2023 at 6:51

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This is "unfortunate" because if you want to use a particle accelerator to probe deeper laws of nature, it needs to accelerate particles to as high energy as possible, not just "any decent fraction of the speed of light." Obviously this is tounge-in-cheek unscientific language; the laws of physics are what they are, and can neither be good nor bad. Because the cyclotron frequency is $$ \omega_c=\frac{q|\mathbf{B}|}{\gamma m_0} $$ Using the description where $m_0$ is the unchanging rest mass, and $\gamma$ is the relativistic lorentz factor. This means that if you accelerate your particles in a cyclotron by applying a constant frequency of changing electric field, you will only be resonantly accelerating the electrons when $\gamma\sim1$. This sets a fundamental limitation on the syncrotron that it can only achieve energies where the particle is moving significantly slower than the speed of light.

The LHC, for example, was able to prove the existnece of the higgs boson by accelerating protons to $7\text{ TeV}$, where $\gamma\sim7000$. This would have been comically impossible for the synchrotron.

However, it only took physicists about 15 years from the development of the cyclotron to invent the synchrocyclotron, where the frequency is slowly decreased, leveraging the phenomenon of autoresonance to accelerate to arbitrarily high energies.

Also, 8 years after the development of the synchrotron, a new design was proposed (the isochronous cyclotron) which used a magnetic field that increased with radius in this paper (The Paths of Ions in the Cyclotron I. Orbits in the Magnetic Field). This design seems to also benefit from focusing its beam, meaning you have longer to accelerate before the beam is lost. I can't find when this design was first built.

So I don't really approve of how much discussion goes into this particular limitation of the synchrotron when it was so easily overcome.

Modern accelerators whose goal is to reach the highest energy to probe new fundamental physics are either linear accelerators or synchrotrons. Synchrotrons are rings whose magnetic field changes to compensate the increase in $\gamma$ as the particles accelerate, meaning you no longer need to make a homogenous magnetic field that covers the entire area of the circle, and you only need to make a thin ring of magnetic field. They also benefit from being able to add other kinds of magnets that focus beams meaning you can accelerate more particles which remain stable longer.

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Your difficulty is about English, not physics. The references to 'this' unfortunate thing and the thing 'which' experimenters want do not point to the same item. Perhaps the author condensed their sentences too much, leaving some links implicit and ending up with a misspeech, and a possible logical break in the mind of a reader who won't gloss over the hitch. Let me rewrite what's being expressed:

  • Punch is what experimenters want.
  • Punch comes from mass-energy.
  • In the relativistic regime, mass-energy grows more than speed in itself does. So increasing speed just a little still gives you punch.
  • However, the cyclotron fails to provide even that because of desynchronization.
  • And that's unfortunate.
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