Why don't we use quater-circular dees instead of semi-circular dees in a Cyclotron

Question

This is the setup, I have in my mind:

O1, O2, O3 and O4 are 4 oscillators.

The arrows in between the Dees represent the alternating EMF the Oscillators will generate.

I think we can easily adjust the frequency of the alternating EMF (T/4) in each oscillator and it seems to work. Any reason it wouldn't? and Why we only use semi-circular Dees?

Why 4 Dees?

Because it would increase the acceleration of the charged particle (not the final velocity); hence, it would take a lot less time to accelerate a particle to a great velocity as compared to that of a cyclotron with 2 Dees. It doesn't seems like a necessary requirement but my main question was is there a reason to use D shaped electrodes?

Answer 1

The vast majority of research cyclotrons don't use the classical Lawrence design anymore. Lawrence's cyclotron was in many ways the simplest circular accelerator you can build, and later designs are much more complex. That said, we can still consider a classical cyclotron with four accelerating gaps.

To review a bit, the cyclotron uses a perpendicular magnetic field to keep accelerated charges within a circular region. As they circulate, the charges cross the gap between the "dees" and experience the electric fields there, which are timed to provide acceleration. For a given particle momentum, the orbit radius is given by the Larmor radius:

$$ r = \frac{p}{|q|B}. $$

Most often the figure of merit that we care about is the energy. In the real world, we are limited by radius (how big we make the cyclotron) and the achievable magnetic field. Clearly, for a given r and B, adding more accelerating gaps will not increase the maximum achievable energy. It will, as you point out, bring a particle up to speed more quickly. Consequently, it will also increase the separation between orbits as the particles spiral outward. For most applications, the amount of time it takes to accelerate particles is not of concern. Orbit separation was also not so important in the early days, since there was often no need to extract beam (just put your target in the path of the spiral). So there was no compelling reason to have anything more complex than two dees.

These days, though, the separation between orbits is very important because we often need to extract a beam of particles from a machine. You want good separation to ensure clean capture and low losses, which reduces radioactive activation of machine components.

There are other issues (relativistic effects, phase/orbit stability) that have led to cyclotron designs that are a far cry from the classical cyclotron. For example, the 590 MeV cyclotron at PSI (which is a "separated-sector" cyclotron) uses four main RF cavities (instead of dees) that fit in between its eight magnets. The K1200 superconducting cyclotron at MSU still has "dees", but their shape is considerably different from a 'D'.