The operation begins when the ions are introduced into the region between two accelerating electrodes, or "dees."[2] Because the ions carry a positive electric charge, they are attracted toward that dee which is electrically negative at the moment. Were it not for the magnetic field, the ions would be accelerated in a straight line; instead they are deflected into a circular path back toward the dee gap. By the time the ions again reach the dee gap, the sign of the electric potential on the dees is reversed, so that now the ions are attracted toward the opposite dee.
As this process of alternating the electric potential is repeated, the ions gain speed and energy with each revolution. This causes them to spiral outward. Finally they strike a target inserted into their path or are extracted from the cyclotron for use as an external beam.
The time required for an ion to complete one loop remains constant as it spirals outward. This is because its velocity increases sufficiently to make up for the increased distance it travels during each turn. This means that the electric potential applied to the dees must alternate at a constant frequency, called the "resonant frequency."
The resonant frequency f is given by the relationship
| f = | He2 π m c | (1) |
where H, e, π, c, and m are constants. H is the strength of the magnetic field of the cyclotron, e is the electric charge carried by the ion, π equals 3.14, c is a conversion factor, and m is the mass of the ion. For example, the resonant frequency for protons accelerated in a 15,000-gauss magnetic field is 23.7 megacycles (Mc).[3] We call such a rapidly alternating potential a "radiofrequency voltage" and the electronic circuit for producing it a "radiofrequency oscillator."
The energy E of an ion emerging from the cyclotron is given by
| E = | H2 R22 | e2mc2 | (2) |
where H, e, and m are as defined above, and R is the radius at which the beam is extracted. From this equation we see that for a given type of ion (where e and m are constant), the energy depends on the diameter and strength of the magnet, but not directly upon the voltage applied to the dees.
The number of revolutions that an ion can make in a conventional cyclotron is limited to about 70 to 100. This is due to a very curious effect: as an ion is accelerated, its mass increases! [This phenomenon is explained by Einstein's special theory of relativity (see Fig. 3).] Referring back to Eq. (1), we see that if the ion mass (m) does not remain constant, but rather increases, then the resonant frequency (f) decreases. But since the dee potential continues alternating at a constant frequency, an ion soon begins to arrive "late" at the dee gap. By the time it has made about 70 to 100 turns an ion is so badly out of phase that it is no longer accelerated.