46 Nuclear Medicine Physics
from the cyclotron. Further, it sets the dimensions (and cost) of almost all the
other main components in the accelerator, clearly showing the importance
of studying axial oscillations around the equilibrium orbit in the cyclotron
construction project. This study is only complete when the acceleration in the
gap between the Ds is taken into account, leading to the electrostatic focusing
effect, which will be discussed in the next section. As far as radial oscillations
are concerned, the cyclotron geometry itself makes the increase in amplitude
resulting from the increasing magnetic field radial value less critical. Nev-
ertheless, the interaction between the magnetic field radial profile and the
evolution of oscillation along the acceleration process is interesting. At the
start of acceleration, when the n value is almost zero, the radial oscillation
frequency is almost equal to the ion orbital frequency. One orbit displaced
in relation to the cyclotron geometric center (and, therefore, to the magnetic
field) will continue to expand in semicircles, increasing the radius around
the same deviation point from the center. However, as the n value grows, the
radial oscillation frequency decreases, resulting in a precession motion such
that the maximum amplitude azimuth moves ahead around the circle. Ions
with essentially the same energy will also present azimuthal angular scat-
tering related to the equilibrium orbit, in addition to radial scattering that is
directly related to the amplitude of free radial oscillation.
2.2.4 Electric Focusing
2.2.4.1 Introduction
A complete study of accelerating ion motion in a cyclotron, especially the
axial components of the respective path, must consider the accelerating gap
between the Ds where they are under the influence of an accelerator electrical
field. This field exerts a force over the ions that divert from the ideal equi-
librium orbit, depending on the shape of the electric field and the radio
frequency cycle phase value when they cross the acceleration gap. We can
find the explanation for this by considering the gaps between the D faces as
an electrical bidimensional lens [69], disregarding the magnetic field action
over the accelerating ions. This approach is correct, because the magnetic
field can be considered uniform along the short distance between the Ds. The
approach remains valid for the first revolutions, when crossing the acceler-
ation gap is still a considerable part of the ion path, as this happens in the
central region of the cyclotron where the magnetic field is almost uniform,
ensuring only circular motion. This is why, when tracing the accelerating ion
path on a plane, we can consider the electrical field lines of force between the
Ds at a given moment as a function of the axial coordinate z and a coordi-
nate s, representing the motion direction ordinate when passing through the
accelerating region (Figure 2.12).
When moving in the plane in which the axial coordinate is zero, the ion only
experiences the action of the electrical field component along the s coordinate,
responsiblefor its energy increase. When its axial coordinate is not zero, it also