28
E. Wilson and B. J. Holzer
A particle arriving at a quadrupole lens at a displacement x obeys Hill’s equation
x
+ kx = 0.
(2.33)
Hence the small deflection θ is just:
Δx
= −kxx.
(2.34)
Comparing quadrupoles with optical lenses we remember that = 1/f and is
the power of the lens and that the matrix, for a thin lens, can be written:
1 0
− kk 1
.
(2.35)
Under the influence of these focusing and defocusing fields, a particle trajectory
will finally look like a more or less zig-zag shaped curve; which for the example of
eight regular cells it is shown in Fig. 2.9.
Quadrupoles are sometimes not short compared to their focal length. One must
therefore use the matrices for a long quadrupole when one comes to compute the
final machine:
M F =
cos
√
k
1
√
k
sin
√
k
−
√
k sin
√
k cos
√
k
, and
M D =
cosh
√
k
1
√
k
sinh
√
k
−
√
k sinh
√
k cosh
√
k
.
(2.36)
x(mm)
s(m)
10
-10
0
1 0
20
30
40
Fig. 2.9 A single particle trajectory in a ring: At each part of the lattice the amplitude and angle,
(x, x ) of the particle are described by a matrix transformation, according to Eq. (2.32). The blue
line corresponds to an ideal particle, with x = x = 0 and so refers to the ideal orbit