Radiofequency pillbox-like cavities are used to accelerate particles, as shown here. They act grossly speaking like a LC circuit, so they are designed to work with a specific frequency.
This already entails a limit on the number of joint cavities you can construct, since the frequency of the alternating electric field will no longer catch up with the accelerating particle as it passes through each unit.
But… after a revolution along the accelerator, the same particle will be travelling at a much higher speed, so in order to give it more energy you should increase the frequency of the RF cavities, which can't be done.
Still, particles at LHC come across these cavities almost $11000$ times per second and are able to gain up energy up to $6.5\,\rm TeV$ (protons). How is that possible?
Best Answer
The fact that speed increases along the path means the number of synchronized cavities that can be connected closely together into an effective accelerating device is limited, yes.
No. This RF frequency is related to frequency of orbital motion of the particle in the ring (in the simplest case, it's the same value). For particle speeds much lower than the speed of light, this orbital frequency is given by the so-called cyclotron frequency $$ \omega_c = \frac{qB}{m} $$ which is constant, because it depends on mass, charge and magnetic field, but does not depend on particle speed.
To achieve high speeds close to speed of light, a cyclotron with acceleration cavities operating at this constant frequency won't work, because due to relativity, the real orbital frequency becomes dependent on the particle speed. That's why synchrotrons were developed, which adapt the frequency of RF oscillations in cavities to the real orbital frequency of the particles, so that the electric field acts on the particles in the right phase to accelerate them efficiently. As the particle speeds up, due to special relativity, this frequency becomes more and more lower than the cyclotron frequency $\omega_c$.