4 Impedance and Collective Effects
155
the dynamic aperture are eventually lost. The dynamic aperture is usually evaluated
by tracking particles with a computer program through the machine where they
experience the fields from the machine elements and other effects such as wake
fields or the beam–beam interaction.
Since the beam–beam interaction is basically a very non-linear lens in the
machine, it distorts the optical properties and it may create a noticeable beating
of the β-function around the whole machine and at the location of the beam–
beam interaction itself. This can be approximated by inserting a quadrupole which
produces the same tune shift at the position of the beam–beam interaction. The
r.m.s. beam size at the collision point is now proportional to
β ∗
p , where β ∗
p is the
perturbed β-function which can be significantly different from the unperturbed βfunction β ∗ . This in turn changes the strength of the beam–beam interaction and the
parameters have to be found in a self-consistent form. This is called the dynamic
beta effect. This is a first deviation from our assumption that the beams are static
non-linear lenses. A strong dynamic beta effect was found in LEP [161] due to its
very large tune shift parameters.
Another effect that can be observed in particular in e + e − colliders is the blow
up of the emittance which naturally limits the reachable beam–beam tune shifts.
4.6.3.4 Beam–Beam Limit
In e + e − colliders the beam sizes are usually an equilibrium between the damping
due to the synchrotron radiation and heating mechanisms such as quantum excitation, intra-beam scattering and very importantly, the beam–beam effect. This leads
to a behaviour that is not observed in a hadron collider. When the luminosity is
plotted as a function of the beam intensity, it should increase approximately as the
current squared [162], in agreement with
L =
N 2 kf
4πσ x σ y
(4.62)
Here k is the number of bunches per beam and f the revolution frequency [162].
At the same time the beam–beam parameter ξ should increase linearly with the
beam intensity according to (4.60)
ξ y =
Nr e β y
2πγ σ y
σ x + σ y
(4.63)
In all e + e − colliders the observation can be made that above a certain current,
the luminosity increases approximately proportional to the current, or at least much
less than with the second power. Another observation is that at the same value
of the intensity the beam–beam parameter ξ saturates. This limiting value of ξ is
commonly known as the beam–beam limit.
155
the dynamic aperture are eventually lost. The dynamic aperture is usually evaluated
by tracking particles with a computer program through the machine where they
experience the fields from the machine elements and other effects such as wake
fields or the beam–beam interaction.
Since the beam–beam interaction is basically a very non-linear lens in the
machine, it distorts the optical properties and it may create a noticeable beating
of the β-function around the whole machine and at the location of the beam–
beam interaction itself. This can be approximated by inserting a quadrupole which
produces the same tune shift at the position of the beam–beam interaction. The
r.m.s. beam size at the collision point is now proportional to
β ∗
p , where β ∗
p is the
perturbed β-function which can be significantly different from the unperturbed βfunction β ∗ . This in turn changes the strength of the beam–beam interaction and the
parameters have to be found in a self-consistent form. This is called the dynamic
beta effect. This is a first deviation from our assumption that the beams are static
non-linear lenses. A strong dynamic beta effect was found in LEP [161] due to its
very large tune shift parameters.
Another effect that can be observed in particular in e + e − colliders is the blow
up of the emittance which naturally limits the reachable beam–beam tune shifts.
4.6.3.4 Beam–Beam Limit
In e + e − colliders the beam sizes are usually an equilibrium between the damping
due to the synchrotron radiation and heating mechanisms such as quantum excitation, intra-beam scattering and very importantly, the beam–beam effect. This leads
to a behaviour that is not observed in a hadron collider. When the luminosity is
plotted as a function of the beam intensity, it should increase approximately as the
current squared [162], in agreement with
L =
N 2 kf
4πσ x σ y
(4.62)
Here k is the number of bunches per beam and f the revolution frequency [162].
At the same time the beam–beam parameter ξ should increase linearly with the
beam intensity according to (4.60)
ξ y =
Nr e β y
2πγ σ y
σ x + σ y
(4.63)
In all e + e − colliders the observation can be made that above a certain current,
the luminosity increases approximately proportional to the current, or at least much
less than with the second power. Another observation is that at the same value
of the intensity the beam–beam parameter ξ saturates. This limiting value of ξ is
commonly known as the beam–beam limit.