Theory, Design and Tests on a Prototype Module of a Compact ...

8 2. LINAC AND SCL ACCELERATORS
parameters involved; also a short review on LIBO particles dynamic
<strong>and</strong> on mechanical aspects is presented.
1. Fundamental definitions
To accelerate particles we use an electric field; this properties is
represented through the Lorentz Force equation
F = q( E + v × B)
which gives the force acting on a point charge q in the presence of
electromagnetic field.
The first idea to have particles accelerated by a longitudinal electric
field is to use a circular waveguide operating in the mode T M01, but
in this case the phase velocity of the field is greater than the velocity
of light <strong>and</strong> the particles would never have synchronism with the
electromagnetic wave, <strong>and</strong> therefore no continuous acceleration.
Then, we need to lower the phase velocity. One way could be to
charge the waveguide with equally spaced disks, let us call L the distance
between disks. The study of such a structure is based on two
fundamental points:
• Floquet’s Theorem: In a lossless spatial periodic structure, the
wave function is periodic too <strong>and</strong> it may differ from a period
to the next only by a factor like e −jkL .
• The boundary conditions cannot be satisfied by a single mode
of the structure <strong>and</strong> there is a continuous spectrum of space
harmonics (Fourier series).
Figure 2.1. Dispersion diagram of a periodic structure
(loaded waveguide); unloaded cavity case is shown for
comparison.
Interesting features are observable in figure 2.1, which shows the
dispersion diagram of an infinite periodic structure <strong>and</strong>, for comparison,
the diagram of an infinite wide waveguide:
• There is a limited passb<strong>and</strong> effect between ωc <strong>and</strong> ωπ. The
frequency range of passb<strong>and</strong> is related to the coupling between
cavities.