The Linearization of the Equations of Motion
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FIGURE 4.1: A homogeneous magnetic dipole.
4.3 Deflectors
4.3.1 The Homogeneous Magnetic Dipole
The next particle optical element we want to study is the magnetic dipole,
consisting of a homogeneous, hence constant, magnetic field in the y-direction.
We consider that the dipole element acts to bend the reference orbit by the
bending angle φ with the bending radius R 0 , so the curvature is h = 1/R 0 .
We also note that from eq. (4.1), h = B y0 /χ m0 . In terms of the quantities
describing the linearized fields, we have
B y0 = constant, n b = 0,
while
E = 0. Keeping in mind magnet design, such a field can be obtained
very schematically as shown in Fig. 4.1.
Let us now consider the equations of motion; we obtain
x
= a,
a
= −h
2 x + h
1 + η 0
2 + η 0
δ,
y
= b,
b
= 0,
l
= −h
1 + η 0
2 + η 0
x +
1
(2 + η 0 )
2 δ,
δ
= 0.
First we observe that if we choose h = 0, we obtain a
= 0, and we have the
same situation as in the case of a drift. But even for the case of h = 0, the
motion of the y-direction behaves simply like a drift, and we always have
y f = y i + b i L,
b f = b i ,
where L is the arc length of the reference orbit in the dipole and L = R 0 φ.