246
An Introduction to Beam Physics
T
l/β 0 λ
0.5
1.0
1.5
2.0
0
0.5
1.0
FIGURE 10.2: Dependence of transit time factor on length of the cavity.
particle may not move along the reference orbit, which is a closed curve,
the effect of the change of arrival time due to the transverse motion tends
to average out due to betatron oscillation. This will become clearer below
when the large difference between the frequencies of the transverse oscillations
and that of the longitudinal one is shown. As a result, we only consider a
particle that moves along the close orbit of its energy. Let us consider a ring
with midplane symmetry that uses pure magnetic elements for bending and
transverse focusing. Hence eq. (3.17) can be simplified as
l
=
(1 + hx δ )
1 + η
1 + η 0
p 0
p s (δ)
− 1
κ
v 0
=
(1 + hx δ )
1 + η
1 + η 0
η (2 + η)
η 0 (2 + η 0 )
− a 2
δ − 1
κ
v 0
=
(1 + hx δ )
1 +
η 0
1 + η 0
δ
1 + 2
1 + η 0
2 + η 0
δ +
η 0
2 + η 0
δ 2 − a 2
δ − 1
κ
v 0
,
where x δ and a δ are the horizontal position and momentum of the closed
orbit of an off-momentum particle. The difference in arrival time between the
reference particle is
Δt =
1
v 0
C
0
(1+ hx δ )
1+
η 0
1 + η 0
δ
1+ 2
1 + η 0
2 + η 0
δ +
η 0
2 + η 0
δ 2 − a 2
δ − 1
ds.