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

16 2. LINAC AND SCL ACCELERATORS
f [GHz]
3.08
3.06
3.04
3.02
3
2.98
2.96
2.94
f
ac
f
cc
2.92
0 0.2 0.4 0.6 0.8 1
φ/π
f [GHz]
3.1
3.08
3.06
3.04
3.02
3
2.98
2.96
f
ac
f
cc
2.94
0 0.2 0.4 0.6 0.8 1
φ/π
Figure 2.9. The dispersion diagrams in the case of a
biperiodic structure with fa = 3 GHz, fc = 3 GHz,
k = 0.04, k2 = 0.005 <strong>and</strong> k3 = 0 on the left <strong>and</strong>
fc = fa/ √ 1 − k2 on the right.
fa; the total number of cavities is therefore 4N + 1. The coupling
between accelerating <strong>and</strong> coupling cavities is k, between non-adjacent
accelerating cells is k2, <strong>and</strong> between coupling cells is k3. With the
hypothesis of infinite quality factor, the dispersion formula is [14]
k 2 cos 2
φ = 1 − f 2 a
f 2
+ k2 cos 2φ 1 −
q
f 2 c
f 2
+ k3 cos 2φ , (2.8)
q
where φ is the phase advance. In the previous section we saw that
for a finite structure the phase advance can assume only a finite set
of value, the number of which is equal to the number of resonators.
The previous relation (2.8) shows that there are two solution for the
φ = π/2 frequency:
f ac
π/2 =
fa
√ , f
1 − k2
cc
π/2 =
fc
√ ,
1 − k3
that deals to a forbidden range of frequency, which is the stop-b<strong>and</strong>, as
it is shown in figure 2.9 (left). If k2 = 0, then is sufficient to choose
fa = fc, where if k2 = 0 one has to choose fc = fa/ √ 1 − k2, the result
of this second case is shown in the figure 2.9 (right).
4. LIBO Specifications
LIBO is a 3 GHz side coupled proton linac. It was conceived as a
LInac BOoster 5 , intended to be mounted downstream of a cyclotron
of about 60 MeV (of which many exist in hospitals <strong>and</strong> physics laboratory),
with the aim to boost the energy of the proton beam up to
200 MeV, an energy needed for therapy of deep seated tumors. All the
matters concerning this section are reported in [22, 23, 24, 25].
The beam intensity needed for protontherapy is relatively small,
about 2 · 10 10 particles per second, i.e. a few nanoamperes, while the
5 This is the origin of word LIBO.