26 unifying physics of accelerators, lasers and plasma
2.3.4 Linear betatron equations of motion
The particle trajectory in an accelerator is initially defined
by dipole magnets, therefore a curvilinear coordinate system is
y
best used to describe the motion of the particles; see Fig. 2.6.
A reference orbit is usually selected, corresponding to an ideal
particle, which typically has a nominal energy and zero transverse offsets and angles.
The focusing elements, quadrupoles and higher-order elx
s
ements are placed in space so that their centers correspond to
the reference orbit.
FIGURE 2.6
The motion of a charged particle with respect to the referFrenet–Serret curvilinear co ence orbit and along the curvilinear abscissa (s in Fig. 2.6), and
ordinate system.
influenced by the the magnetic fields of dipole magnets and
quadrupole magnets, is given by the linear Hill’s equations
d 2 y + K y (s)y = 0
(2.18)
ds 2
where transverse coordinate y stands for either a horizontal
or vertical axis (y = x, z).
Let’s write these equations down for the horizontal
1
1 ∂B
K x ( ) =
−
z (s)
s
(2.19)
ρ 2 (s) Bρ ∂x
and the vertical
1 ∂B
(
z (s)
K z s) =
(2.20)
Bρ ∂x
planes. We see that the equation almost exactly resembles
those derived in the previous section (see Eq. 2.16 and
FIGURE 2.7
Eq. 2.17) except for an additional term 1/ρ 2 (s), which correShifted circles cross.
sponds to a weak focusing of a dipole.
The origin of the term corresponding to the weak focusing
in a dipole can be illustrated by the following example. Consider the shifted circles in Fig. 2.7. They cross in two points
(we will ignore second-order effects). Translating this example of shifted circles to a dipole magnet whose field fills a
half plane as shown in Fig. 2.8, we conclude by observation
%
that the trajectories of the particles in this dipole exhibit an
equivalent “focusing” with the wavelength of motion (along
FIGURE 2.8
the curvilinear coordinate s) given by 2 πρ, corresponding to
Illustration of the origin of
weak focusing in dipoles.
x = x 0 sin (s/ρ)
or to the following equation
d 2 x x
+
= 0
(2.21)
ds 2 ρ 2
which has the same term 1/ρ 2 (s) as the Hill’s equations above.
The above derivations generally assumed no periodicity;
however, in a circular machine, K x ,K z and ρ are periodic.
These are linear equations and can be integrated, which will
be discussed in the next section.
2.3.4 Linear betatron equations of motion
The particle trajectory in an accelerator is initially defined
by dipole magnets, therefore a curvilinear coordinate system is
y
best used to describe the motion of the particles; see Fig. 2.6.
A reference orbit is usually selected, corresponding to an ideal
particle, which typically has a nominal energy and zero transverse offsets and angles.
The focusing elements, quadrupoles and higher-order elx
s
ements are placed in space so that their centers correspond to
the reference orbit.
FIGURE 2.6
The motion of a charged particle with respect to the referFrenet–Serret curvilinear co ence orbit and along the curvilinear abscissa (s in Fig. 2.6), and
ordinate system.
influenced by the the magnetic fields of dipole magnets and
quadrupole magnets, is given by the linear Hill’s equations
d 2 y + K y (s)y = 0
(2.18)
ds 2
where transverse coordinate y stands for either a horizontal
or vertical axis (y = x, z).
Let’s write these equations down for the horizontal
1
1 ∂B
K x ( ) =
−
z (s)
s
(2.19)
ρ 2 (s) Bρ ∂x
and the vertical
1 ∂B
(
z (s)
K z s) =
(2.20)
Bρ ∂x
planes. We see that the equation almost exactly resembles
those derived in the previous section (see Eq. 2.16 and
FIGURE 2.7
Eq. 2.17) except for an additional term 1/ρ 2 (s), which correShifted circles cross.
sponds to a weak focusing of a dipole.
The origin of the term corresponding to the weak focusing
in a dipole can be illustrated by the following example. Consider the shifted circles in Fig. 2.7. They cross in two points
(we will ignore second-order effects). Translating this example of shifted circles to a dipole magnet whose field fills a
half plane as shown in Fig. 2.8, we conclude by observation
%
that the trajectories of the particles in this dipole exhibit an
equivalent “focusing” with the wavelength of motion (along
FIGURE 2.8
the curvilinear coordinate s) given by 2 πρ, corresponding to
Illustration of the origin of
weak focusing in dipoles.
x = x 0 sin (s/ρ)
or to the following equation
d 2 x x
+
= 0
(2.21)
ds 2 ρ 2
which has the same term 1/ρ 2 (s) as the Hill’s equations above.
The above derivations generally assumed no periodicity;
however, in a circular machine, K x ,K z and ρ are periodic.
These are linear equations and can be integrated, which will
be discussed in the next section.