e 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

He
f = --------- , (1)
2[pi]mc

where H, e, [pi], 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, [pi] 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

H^2 R^2 e^2
E = ------- ---- , (2)
2 mc^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 i