How does the cyclotron work

Functional model of a cyclotron (classic)

Construction:

A cyclotron consists of two hollow, semicircular metal electrodes, the so-called Duanten. A high-frequency alternating voltage is connected to the duants, which alternately charges the duants positively and negatively. The opposing charge of the two elements creates a strong electric field in the gap between the elements. Electromagnets are located above and below the elements, which generate a constant, homogeneous magnetic field in the elements. In addition, there is a particle source inside a Duanten that emits free ions.

Function:

The particle source T emits, for example, protons (positively charged) at a speed $ v_0 $ into the gap between the quanta. If Duant 1 has a negative charge, it is accelerated by the E-field in the gap. If the particle enters the quantum 1, it leaves the E-field (each quantum acts as a Faraday cage). The Duant is penetrated by the B-field of the electromagnet. Therefore, a Lorentz force now acts on the proton, which forces it onto a circular path. While the proton is in Duant 1, the polarity of the Duants is reversed - the E-field changes its direction. After passing through a semicircle, the proton leaves the Duant again and enters the E-field, which is now appropriately aligned again. It is accelerated again until it enters Duant 2. Here again a Lorentz force acts, which leads to a circular path. However, due to the now higher speed of the proton, the radius of this orbit is larger than the radius in Duanten 1 was. Longer distances and higher speeds mean that the throughput time through a Duanten remains constant.
This process is repeated a few times, with the particle getting faster and faster and the radius of the circular path increasing. At the end, the particle is deflected out of the Duanten by an additional E-field of a deflection electrode and can be shot at any target.

Restrictions:

A cyclotron of this type is only suitable for particle accelerations up to approx. $ 0 {,} 1 \ cdot c $, since from here on the relativistic increase in mass ensures that the period of rotation $ T $ is no longer constant, but increases. As a result, it gets "out of step" with the constant external alternating voltage, which leads to an acceleration that is no longer optimal.
For this reason, cyclotrons are usually used to accelerate heavy particles such as protons or deuterons. Due to their high mass, a relatively large amount of energy must be supplied to them before they reach critical speeds. Light particles such as electrons reach the critical speed after just a few revolutions.
Protons in the classic cyclotron with acceleration voltages of a few hundred volts reach energies of around 10 MeV after around 50 cycles.