6 Design and Principles of Synchrotrons and Circular Colliders
251
given by
τ q,u =
1
2
τ u
e ξ u
ξ u
, where ξ u =
A 2
u
2σ 2
u
, for u = x, y, z, ξ u 20,
(6.60)
where the τ u are the radiation damping times and A u are appropriate acceptances
[53, 57, 58]. For the synchrotron mode, A z is the RF bucket half-height
A z
E 0
2
=
2U 0
π |η| h RF E 0
(eV RF /U 0 )
2
− 1 − arccos (U 0 /eV RF )
.
(6.61)
For adequate lifetime at small intensity, the mechanical and dynamic apertures
and RF voltage must be large enough.
The bunch length is given by σ z = c|η|σ E /(ω s E 0 ) where ω s is the angular
synchrotron frequency and η α c the frequency slip factor ([53], Sects. 2.5.2 and
2.5.3).
6.7.2 Design of Colliders
Colliders are designed from the interaction point outwards. The classical design is
based on head-on collisions of flat beams. However a number of other configurations
have been explored and the most promising among them is described in the
following section. In the classical scheme, luminosity (Sect. 6.4) is maximised
by achieving very flat beams, κ = ε y /ε x 1; we consider only beams of equal
energy, size and single bunch population, N b , colliding head-on, with σ ∗
y β ∗
y ;
for generalisations see [53]. The beam-beam effect (Sect. 4.6.1) generally imposes
maximum attainable values on the horizontal and vertical beam-beam parameters
ξ x,y =
r e N b β ∗
x,y
2π
E 0 /mc 2
σ ∗
x,y
σ ∗
x + σ ∗
y
,
(6.62)
where r e is the electron classical radius, β ∗
x,y and σ ∗
x,y,z are the optical functions and
beam sizes at the collision point. Typically, one finds maxξ y = 0.03 − 0.1 with the
highest values attained when the machine is very well corrected (favourable tunes,
central orbits close to design, minimised vertical dispersion) and when radiation
damping is strong. Then the luminosity (Sect. 6.4) can be expressed as
L =
f c N b
2r e
E 0
mc 2
(1 + κ) ξ y
β ∗
y
,
(6.63)
where f c is the frequency at which identical bunches collide; in the simplest case
f c = k b f 0 where k b is the number of bunches per beam.
251
given by
τ q,u =
1
2
τ u
e ξ u
ξ u
, where ξ u =
A 2
u
2σ 2
u
, for u = x, y, z, ξ u 20,
(6.60)
where the τ u are the radiation damping times and A u are appropriate acceptances
[53, 57, 58]. For the synchrotron mode, A z is the RF bucket half-height
A z
E 0
2
=
2U 0
π |η| h RF E 0
(eV RF /U 0 )
2
− 1 − arccos (U 0 /eV RF )
.
(6.61)
For adequate lifetime at small intensity, the mechanical and dynamic apertures
and RF voltage must be large enough.
The bunch length is given by σ z = c|η|σ E /(ω s E 0 ) where ω s is the angular
synchrotron frequency and η α c the frequency slip factor ([53], Sects. 2.5.2 and
2.5.3).
6.7.2 Design of Colliders
Colliders are designed from the interaction point outwards. The classical design is
based on head-on collisions of flat beams. However a number of other configurations
have been explored and the most promising among them is described in the
following section. In the classical scheme, luminosity (Sect. 6.4) is maximised
by achieving very flat beams, κ = ε y /ε x 1; we consider only beams of equal
energy, size and single bunch population, N b , colliding head-on, with σ ∗
y β ∗
y ;
for generalisations see [53]. The beam-beam effect (Sect. 4.6.1) generally imposes
maximum attainable values on the horizontal and vertical beam-beam parameters
ξ x,y =
r e N b β ∗
x,y
2π
E 0 /mc 2
σ ∗
x,y
σ ∗
x + σ ∗
y
,
(6.62)
where r e is the electron classical radius, β ∗
x,y and σ ∗
x,y,z are the optical functions and
beam sizes at the collision point. Typically, one finds maxξ y = 0.03 − 0.1 with the
highest values attained when the machine is very well corrected (favourable tunes,
central orbits close to design, minimised vertical dispersion) and when radiation
damping is strong. Then the luminosity (Sect. 6.4) can be expressed as
L =
f c N b
2r e
E 0
mc 2
(1 + κ) ξ y
β ∗
y
,
(6.63)
where f c is the frequency at which identical bunches collide; in the simplest case
f c = k b f 0 where k b is the number of bunches per beam.