3 Non-linear Dynamics in Accelerators
55
3.3.1.1 Unwanted Non-linear Machine Elements
The largest fraction of machine elements are either dipole or quadrupole magnets.
In the ideal case, these types of magnets have pure dipolar or quadrupolar fields
and behave approximately as linear machine elements. Any systematic or random
deviation from this linear field introduces non-linear fields into the machine lattice.
These effects can dominate the aperture required and limit the stable region of the
beam. The definition of tolerances on these imperfections is an important part of
any accelerator design.
Normally magnets are long enough that a 2-dimensional field representation is
sufficient. The components of the magnetic field can be derived from the potential
and in cylindrical coordinates (r, ,, s = 0) can be written as:
B r (r, ,) =
∞
n=1
(B n sin(nn) + A n cos(nn))
r
R ref
n−1
,
(3.6)
B (r, ,) =
∞
n=1
(B n cos(nn) − A n sin(nn))
r
R ref
n−1
,
(3.7)
where R ref is a reference radius and B n and A n are constants. Written in Cartesian
coordinates we have:
B(z) =
∞
n=1
(B n + iA n )
r
R ref
n−1
(3.8)
where z = x + iy = re ii . The terms n correspond to 2n-pole magnets and the
B n and A n are the normal and skew multipole coefficients. The beam dynamics set
limits on the allowed multipole components of the installed magnets.
3.3.1.2 Wanted Non-linear Machine Elements
In most accelerators the momentum dependent focusing of the lattice (chromaticity)
needs to be corrected with sextupoles [3, 4]. Sextupoles introduce non-linear fields
into the lattice that are larger than the intrinsic non-linearities of the so-called linear
elements (dipoles and quadrupoles). In a strictly periodic machine the correction can
be done close to the origin and the required sextupole strengths can be kept small.
For colliding beam accelerators usually special insertions are foreseen to host the
experiments where the dispersion is kept small and the β-function is reduced to a
minimum. The required sextupole correction is strong and can lead to a reduction
of the dynamic aperture, i.e. the region of stability of the beam. In most accelerators
the sextupoles are the dominant source of non-linearity. To minimize this effect is
an important issue in any design of an accelerator.
55
3.3.1.1 Unwanted Non-linear Machine Elements
The largest fraction of machine elements are either dipole or quadrupole magnets.
In the ideal case, these types of magnets have pure dipolar or quadrupolar fields
and behave approximately as linear machine elements. Any systematic or random
deviation from this linear field introduces non-linear fields into the machine lattice.
These effects can dominate the aperture required and limit the stable region of the
beam. The definition of tolerances on these imperfections is an important part of
any accelerator design.
Normally magnets are long enough that a 2-dimensional field representation is
sufficient. The components of the magnetic field can be derived from the potential
and in cylindrical coordinates (r, ,, s = 0) can be written as:
B r (r, ,) =
∞
n=1
(B n sin(nn) + A n cos(nn))
r
R ref
n−1
,
(3.6)
B (r, ,) =
∞
n=1
(B n cos(nn) − A n sin(nn))
r
R ref
n−1
,
(3.7)
where R ref is a reference radius and B n and A n are constants. Written in Cartesian
coordinates we have:
B(z) =
∞
n=1
(B n + iA n )
r
R ref
n−1
(3.8)
where z = x + iy = re ii . The terms n correspond to 2n-pole magnets and the
B n and A n are the normal and skew multipole coefficients. The beam dynamics set
limits on the allowed multipole components of the installed magnets.
3.3.1.2 Wanted Non-linear Machine Elements
In most accelerators the momentum dependent focusing of the lattice (chromaticity)
needs to be corrected with sextupoles [3, 4]. Sextupoles introduce non-linear fields
into the lattice that are larger than the intrinsic non-linearities of the so-called linear
elements (dipoles and quadrupoles). In a strictly periodic machine the correction can
be done close to the origin and the required sextupole strengths can be kept small.
For colliding beam accelerators usually special insertions are foreseen to host the
experiments where the dispersion is kept small and the β-function is reduced to a
minimum. The required sextupole correction is strong and can lead to a reduction
of the dynamic aperture, i.e. the region of stability of the beam. In most accelerators
the sextupoles are the dominant source of non-linearity. To minimize this effect is
an important issue in any design of an accelerator.