Ring type accelerator solved the difficulties by linear type.

1. the particle circulate inside the accelerator, so, it can be accelerate infinite time in principle.
2. The space require is smaller compare with similar energy output Linac.

However, there is a draw back is, for charged particle running in a circular path, it will radiate energy by EM wave due to the centripetal acceleration, thus, even it is just running in constant speed, it will radiate and energy lost. this is called synchrotron radiation.

There are mainly 2 types of ring accelerators, 1) cyclotron, 2) synchrotron.

Cyclotron

cyclotron is the simplest type, it has 2 D shape cavities and the 2 D shape formed a circle. the 2 D shape cavities is under a magnetic field to blend the particle. and the 2 Ds have different electric potential. when a particle pass from 1 D to the others, due to the potential different, it will be accelerate.

as you can imagine, the potential of the Ds has to be oscillating so that the particle is accelerated when passing each gap between the 2Ds. That frequency is called cyclotron frequency. and it also reflects the particle circulating frequency. surprisingly, the cyclotron frequency only depends on the magnetic field strength, the charge and the mass of the particle.

$\omega = 2\pi f = \frac { B q} {m}$

which means, no matter the particle position, it moves in same frequency. Thus, the outer particle move faster then the inner one.

now a day, cyclotron may not just contains 2 D cavities but any $\frac { 2 \pi }{n}$ cavities. where n is number of cavities. thus, 2Ds is also called π – cavity.

The typical speed it can reach is about 10% of speed of light.

The only draw back is, the energy it can reach is limited, if using fixed B field or E field, due to relativistic effect. (i.e. the cyclotron frequency also depend on the speed ) the particle cannot match the frequency and accelerated, after it goes to relativistic speed.

another factor is the B field strength is limited, even using super conductive magnetic. and the limit of B field, limited the max output.

particle is released at the center of the B-field and go outward as it acquire speed. Thus, the limitation of radius also limited the max speed. and also, a large radius means a large B field area, which raise a problem on uniform on the B-field.

So, there are another type of cyclotron, which changing the B field or E field to cope with the changing frequency. such cyclotron is called Synchrocyclotron. but due to the velocity dependent of the frequency, only certain speed of particle can be accelerate, thus, the intensity of the beam is smaller then cyclotron.

Synchrotron

synchrotron can reach a great energy and accelerate particle very close to speed of light.

it uses a lot beam focusing devices and accelerating devices to accelerate the beam in a very large radius. each device is well tuned, and all devices are well synchronized for different particle. Thus it is a very delicates and sophisticated machine.

1. particle can have every high energy
2. high intensity of beam
3. it can have some section only for linear motion with accelerate.
4. it is not limited by the B field. since the narrow of the beam, a higher forcing B field can be applied.

The only factor reduced the power output is the synchrotron radiation. that’s why they build a bigger and bigger one, since the large radius can reduce the radiation lost.