[Physics] Difference between Betatron and Synchrotron

Question

from my current understanding of linear accelerators, both utilise changing magnetic fields to focus and confine accelerating electrons to a constant orbital radius. Then, what are the main differences in operation/functionality between the two? Apologies for my naivety, thank you!

Answer 1

The betatron is rather unique type of accelerator since it accelerates the charged particles exclusively by the use of a magnet via a time change of the magnetical field $\bf{B}= \nabla \times \bf{A}$. The time change of the vector potential $\bf{A}$ related with the magnetic field $\bf{B}$ creates a non-conversative electrical field (here cgs-units are used): $$ \bf{E} = -\frac{1}{c} \frac{\partial \bf{A}}{\partial t}$$

In different words, it is acceleration of charged particles per induction. Simultaneously the magnetical field serves as guiding field, i.e. keeps the charged particles on a circular track. However, it is rather limited since as soon as the saturation of the magnetic material (typically iron) due to hysteresis is reached, the vector potential does not change anymore and the acceleration process can't continue anymore.

In a synchrotron the magnets are exclusively used for the guidance and focussing of the particle beam. The energy, i.e. the acceleration, is provided by a radio-frequency system where the particle "rides" on a rapidly changing electromagnetic field, it gets its energy from a sinusoidal electrical field which is accompanied by a magnetic field orthogonal to it both generated in (typically) several high Q-cavities (made of copper if it is a warm cavity). As the energy of the particles increases, the field strength of the guiding magnets also increases to keep the particles on track, but its contribution to the acceleration process here is actually minimal. With this set-up substantially higher energies can be reached, which mainly depends on how much field gradient MeV/m with the cavity system can be reached. A stronger limit the is actually again given by the magnetic field of the beam guiding magnets which have to keep the beam at high energy on track, but theoretically this limit can almost always circumvented in choosing an accelerator of larger circumference or even make it linear (as long as the building costs it allows).