The final energy of the particles in a Linac is determined by the voltage across the R-F cavities and the total length of the beam pipe (and hence how many cavities the bunch can pass through). In a circular accelerator, or synchrotron, final energy depends on the ring radius and the maximum magnetic field that can be achieved to maintain the bunches of particles in orbit. The magnets used to bend the charged particles around their circular path are dipoles as opposed to the quadrupoles used to focus the beam.
The particles do not travel in ideal circular orbits but wander in both the horizontal and vertical planes. This is called beteron oscillation and is caused by the divergence of the original beam and fact that the bending is not constant due to the spaces between successive dipole magnets. Oscillations known as synchrotron oscillations occur in the direction in which the beam travels when particles get out of step with the synchronous phase. The synchronous phase is the term given to the condition that the increase in the bending fields of the dipole magnets exactly matches the increase in the particles’ momenta due to the R-F cavities.
Particles in a synchrotron typically complete around 105 laps of the ring before attaining their full energy. However, the orbiting does not stop there. Even once focused, relatively few particle collisions will take place each time the bunches meet. Hence, the beams continue to be collided and observed for several hours during an experiment resulting in enormous amounts of data. This is very important because the massive particles we want to study are very rarely produced even when the energy is available but proved a key limiting factor in the early days of particle physics when images of collisions had to be analysed by hand. Today, this is done by computers.
The name synchrotrons is given to circular accelerators after the synchrotron radiation which limits the energy of electron-positron colliders. When an electron accelerates it radiates energy. A particle traveling in a circle is always accelerating towards the circle’s centre. The greater the radius of the circle and the higher the particle’s velocity, the greater the acceleration. Hence, at a certain velocity, all the energy pumped into an electron (or positron) by the R-F cavities around a beam pipe will simply be radiated away due to its centripetal acceleration. The electron’s kinetic energy will remain constant. This is why synchrotrons are built with such large radii. The second major limiting factor is how strong the bending magnetic fields can be made. This applies to protons as well as electrons. As mentioned above, the faster the particles are traveling, the stronger need to be the dipole B-fields to bend them into a circular path. Again, increasing the length of the circular beam pipe serves to reduce the field strength required.