226
An Introduction to Beam Physics
To match a dispersion free region to a FODO cell, we have
⎛
⎜
⎝
cos (2μ) β sin (2μ) d
− sin (2μ) /β cos (2μ) d
0
0
1
⎞
⎟
⎠
⎛
⎜
⎝
0
0
1
⎞
⎟
⎠ =
⎛
⎜
⎝
D max
0
1
⎞
⎟
⎠ ,
which leads to
(2 cos μ + 1) θ 1 + θ 2 =
θ
2 sin
2 (μ/2)
,
(2 cos μ − 1) θ 1 + θ 2 = 0.
The result is
θ 1 =
θ
4 sin
2 (μ/2)
, θ 2 = θ − θ 1 .
So, we have
θ 1 = θ 2 =
θ
2
,
for μ =
π
2
.
θ 1 = θ, θ 2 = 0, for μ =
π
3
.
Dispersion suppressors are widely used in high energy accelerators where
the achromatic straight section constitutes only a small portion of the ring.
Since the main part of the ring is made up of arcs, the cost-effective way to
build such a ring is to pack dipole magnets as close as possible. A FODO cell is
the best choice for this purpose. Another way to save cost is to keep the beam
pipe as small as possible, which saves not only due to smaller pipes themselves,
but also, more importantly, smaller magnets. Dispersion suppressors help to
keep beam size small by keeping the dispersion matched. In fact, there is
another parameter that plays an important role in the optimization process,
which is the length of the FODO cell. Both β max and D max are proportional
to the length of the cell. A shorter cell leads to smaller beam size, but tends to
decrease the packing factor, which is the ratio of the length of total bending
over the total length of the cell.
9.2 Symmetric Achromats
Achromats based on mirror symmetry are widely used in beamlines and
accelerators, especially synchrotron light sources. One difference between a
synchrotron light source and a high energy accelerator is in the number of
experiments it supports. While a high energy accelerator usually supports
around ten fixed target experiments and a handful of collider experiments (less
An Introduction to Beam Physics
To match a dispersion free region to a FODO cell, we have
⎛
⎜
⎝
cos (2μ) β sin (2μ) d
− sin (2μ) /β cos (2μ) d
0
0
1
⎞
⎟
⎠
⎛
⎜
⎝
0
0
1
⎞
⎟
⎠ =
⎛
⎜
⎝
D max
0
1
⎞
⎟
⎠ ,
which leads to
(2 cos μ + 1) θ 1 + θ 2 =
θ
2 sin
2 (μ/2)
,
(2 cos μ − 1) θ 1 + θ 2 = 0.
The result is
θ 1 =
θ
4 sin
2 (μ/2)
, θ 2 = θ − θ 1 .
So, we have
θ 1 = θ 2 =
θ
2
,
for μ =
π
2
.
θ 1 = θ, θ 2 = 0, for μ =
π
3
.
Dispersion suppressors are widely used in high energy accelerators where
the achromatic straight section constitutes only a small portion of the ring.
Since the main part of the ring is made up of arcs, the cost-effective way to
build such a ring is to pack dipole magnets as close as possible. A FODO cell is
the best choice for this purpose. Another way to save cost is to keep the beam
pipe as small as possible, which saves not only due to smaller pipes themselves,
but also, more importantly, smaller magnets. Dispersion suppressors help to
keep beam size small by keeping the dispersion matched. In fact, there is
another parameter that plays an important role in the optimization process,
which is the length of the FODO cell. Both β max and D max are proportional
to the length of the cell. A shorter cell leads to smaller beam size, but tends to
decrease the packing factor, which is the ratio of the length of total bending
over the total length of the cell.
9.2 Symmetric Achromats
Achromats based on mirror symmetry are widely used in beamlines and
accelerators, especially synchrotron light sources. One difference between a
synchrotron light source and a high energy accelerator is in the number of
experiments it supports. While a high energy accelerator usually supports
around ten fixed target experiments and a handful of collider experiments (less