56
W. Herr and E. Forest
Another source of non-linearities can be octupoles used to generate amplitude
dependent detuning to provide Landau damping in case of instabilities.
3.3.2 Beam–Beam Effects and Space Charge
A strong source of non-linearities are the fields generated by the beam itself. They
can cause significant perturbations on the same beam (space charge effects) or on
the opposing beam (beam-beam effects) in the case of a colliding beam facility.
As an example, for the simplest case of round beams with the line density n and
the beam size σ the field components can be written as:
E r = −
ne
4ππ 0
·
∂
∂r
∞
0
exp(−
r 2
(2σ 2 +q)
)
(2σ 2 + q)
dq,
(3.9)
and
B = −
neβcμ 0
4π
·
∂
∂r
∞
0
exp(−
r 2
(2σ 2 +q)
)
(2σ 2 + q)
dq.
(3.10)
In colliding beams with high density and small beams these fields are the dominating
source of non-linearities. The full treatment of beam-beam effects is complicated
due to mutual interactions between the two beams and a self-consistent treatment is
required. in the presence of all other magnets in the ring.
3.4 Map Based Techniques
In the standard approach to single particle dynamics in rings, the equations of
motion are introduced together with an ansatz to solve these equations. In the case
of linear motion, this ansatz is due to Courant-Snyder [2]. However, this treatment
must assume that the motion of a particle in the ring is stable and confined. For
a non-linear system this is a priori not known and the attempt to find a complete
description of the particle motion must fail.
The starting point for the treatment of the linear dynamics in synchrotrons is
based on solving a linear differential equation of the Hill type.
d 2 x(s)
ds 2 +
a 0 + 2
∞
n=1
a n · cos(2ns)
K(s)
x(s) = 0 .
W. Herr and E. Forest
Another source of non-linearities can be octupoles used to generate amplitude
dependent detuning to provide Landau damping in case of instabilities.
3.3.2 Beam–Beam Effects and Space Charge
A strong source of non-linearities are the fields generated by the beam itself. They
can cause significant perturbations on the same beam (space charge effects) or on
the opposing beam (beam-beam effects) in the case of a colliding beam facility.
As an example, for the simplest case of round beams with the line density n and
the beam size σ the field components can be written as:
E r = −
ne
4ππ 0
·
∂
∂r
∞
0
exp(−
r 2
(2σ 2 +q)
)
(2σ 2 + q)
dq,
(3.9)
and
B = −
neβcμ 0
4π
·
∂
∂r
∞
0
exp(−
r 2
(2σ 2 +q)
)
(2σ 2 + q)
dq.
(3.10)
In colliding beams with high density and small beams these fields are the dominating
source of non-linearities. The full treatment of beam-beam effects is complicated
due to mutual interactions between the two beams and a self-consistent treatment is
required. in the presence of all other magnets in the ring.
3.4 Map Based Techniques
In the standard approach to single particle dynamics in rings, the equations of
motion are introduced together with an ansatz to solve these equations. In the case
of linear motion, this ansatz is due to Courant-Snyder [2]. However, this treatment
must assume that the motion of a particle in the ring is stable and confined. For
a non-linear system this is a priori not known and the attempt to find a complete
description of the particle motion must fail.
The starting point for the treatment of the linear dynamics in synchrotrons is
based on solving a linear differential equation of the Hill type.
d 2 x(s)
ds 2 +
a 0 + 2
∞
n=1
a n · cos(2ns)
K(s)
x(s) = 0 .