7 Design and Principles of Linear Accelerators and Colliders
299
7.2 High Luminosity Issues and Beam-Beam Effects
D. Schulte
In linear colliders, the colliding beams have extremely small transverse dimensions
σ x,y to reach high luminosity. Each beam exerts a strong electro-magnetic force
on the other beam, which is focusing in case of electron-positron collisions.
This disruption can shrink the beam size significantly during collision, the socalled pinch effect [25–27]. This increases the luminosity but the bending of the
particles’ trajectories stimulates them to radiate so-called beamstrahlung photons,
a process similar to synchrotron radiation [28–32]. Consequently, not all collisions
take place at the nominal centre-of-mass energy. Hence, one needs to choose the
beam parameters in order to limit the beamstrahlung and to achieve an acceptable
luminosity spectrum for the experiment. High luminosity with low beamstrahlung
is usually achieved by using flat beams σ x σ y ) as shown below. Approximate
formulae are used, since full analytic treatment of the beam-beam interaction is
for most parameters not possible. Simulation codes are used for precise numerical
predictions, in particular CAIN and GUINEA-PIG [33–35].
It should be noted that one usually needs a horizontal crossing angle θ c between
the two beam lines at the interaction point. This separation of incoming and
outgoing beam allows to efficiently extract collision debris and hence to avoid
large beam losses after the collision. For short distances in-between bunches in
each beam pulse, the crossing angle also reduces the impact of parasitic crossings
of incoming and outgoing bunches. The luminosity reduction due to the crossing
angle is avoided by using a so-called crab-crossing scheme. Before the collision
a transverse deflecting cavity introduces a rotation around the vertical axis, such
that the beams are aligned to the longitudinal axis of the laboratory system at the
collision rather than the direction of motion. Hence the two bunches fully overlap
during the collision while moving together horizontally, like crabs. In this case the
crossing angle hardly affects the beam-beam interaction and can be neglected in the
further considerations.
In high energy linear colliders like ILC and CLIC developed in Sect. 7.3, the
luminosity is limited by beamstrahlung, the achievable vertical beam size and the
efficiency of the main linac RF. This is seen by expressing it as a function of the
number of particles per bunch N, the number of bunches per beam pulse n b , the
repetition rate of beam pulses f r and the luminosity enhancement factor H D , which
tends to be in the range of 1–2:
L = H D
N 2
4πσ x σ y
n b f r .
(7.1)
Ignoring the usually small variations of H D , one obtains the simple dependence:
L ∝
N
σ x
1
σ y
ηP wall .
(7.2)
299
7.2 High Luminosity Issues and Beam-Beam Effects
D. Schulte
In linear colliders, the colliding beams have extremely small transverse dimensions
σ x,y to reach high luminosity. Each beam exerts a strong electro-magnetic force
on the other beam, which is focusing in case of electron-positron collisions.
This disruption can shrink the beam size significantly during collision, the socalled pinch effect [25–27]. This increases the luminosity but the bending of the
particles’ trajectories stimulates them to radiate so-called beamstrahlung photons,
a process similar to synchrotron radiation [28–32]. Consequently, not all collisions
take place at the nominal centre-of-mass energy. Hence, one needs to choose the
beam parameters in order to limit the beamstrahlung and to achieve an acceptable
luminosity spectrum for the experiment. High luminosity with low beamstrahlung
is usually achieved by using flat beams σ x σ y ) as shown below. Approximate
formulae are used, since full analytic treatment of the beam-beam interaction is
for most parameters not possible. Simulation codes are used for precise numerical
predictions, in particular CAIN and GUINEA-PIG [33–35].
It should be noted that one usually needs a horizontal crossing angle θ c between
the two beam lines at the interaction point. This separation of incoming and
outgoing beam allows to efficiently extract collision debris and hence to avoid
large beam losses after the collision. For short distances in-between bunches in
each beam pulse, the crossing angle also reduces the impact of parasitic crossings
of incoming and outgoing bunches. The luminosity reduction due to the crossing
angle is avoided by using a so-called crab-crossing scheme. Before the collision
a transverse deflecting cavity introduces a rotation around the vertical axis, such
that the beams are aligned to the longitudinal axis of the laboratory system at the
collision rather than the direction of motion. Hence the two bunches fully overlap
during the collision while moving together horizontally, like crabs. In this case the
crossing angle hardly affects the beam-beam interaction and can be neglected in the
further considerations.
In high energy linear colliders like ILC and CLIC developed in Sect. 7.3, the
luminosity is limited by beamstrahlung, the achievable vertical beam size and the
efficiency of the main linac RF. This is seen by expressing it as a function of the
number of particles per bunch N, the number of bunches per beam pulse n b , the
repetition rate of beam pulses f r and the luminosity enhancement factor H D , which
tends to be in the range of 1–2:
L = H D
N 2
4πσ x σ y
n b f r .
(7.1)
Ignoring the usually small variations of H D , one obtains the simple dependence:
L ∝
N
σ x
1
σ y
ηP wall .
(7.2)