564
M. Lamont et al.
on collisions can be best optimized by deploying round beams (equal beam sizes
and emittances in both planes). The generation of equal betatron functions at the IP
can be best provided by a triplet quadrupole configuration of the focusing elements
next to the IP (this implies equal β ∗ values in both planes for equal peak betatron
functions for both planes inside the triplet magnets). The beam-beam parameter in
this case is given by Eq. (9.74) and is entirely independent from the beta function
values at the IP. Optimizing the instantaneous luminosity in this case therefore
implies:
• increasing the beam brightness (N b // x,y ) until the beam-beam limit is reached;
• increasing the bunch population at constant brightness;
• minimizing the beta functions at the IP in both planes.
The second point is limited by the available aperture along the whole collider and by
collective effects that might limit the maximum bunch intensity. The third point is
limited by the available aperture of the triplet elements next to the IP (the maximum
beta function in these elements is proportional to 1/β ∗ ), the ‘hour glass’ effect
and eventual chromatic aberrations due to large betatron functions in the focusing
elements. The hour glass effect depends strongly on the bunch length and is only
relevant for a configuration with β ∗ ≤ σ s . Shortening the bunch length might be
another strategy for maximizing the luminosity in this case but the minimum bunch
length might itself be limited by other constrains such as the available RF voltage
and limitations on the maximum acceptable Intra-Beam-Scattering (IBS) growth
rates.
The maximum number of bunches in the collider is then either limited by
collective effects (e.g. electron-cloud effect for proton bunches), the maximum
acceptable stored beam power (e.g. machine protection issues) and the rate of
particle production in case the collider uses anti-protons in the collisions (e.g. the
Tevatron).
9.8.5 Luminosity Optimization in the Presence of a Crossing
Angle
If the operation of the collider permits a large number of bunches, it might be
necessary to introduce a crossing angle at the IP in order to avoid unwanted
collisions of the two beams next to the IP (e.g. the KEK-B factory and the LHC).
A crossing angle reduces the luminosity and the beam-beam parameter via the
geometric form factor F in Eq. (9.70). The geometric form factor is a function
of the crossing angle, the bunch length and the transverse beam size in the plane
of the crossing angle and therefore crossing angle operation introduces additional
parameter dependencies which make the luminosity optimization more complex.
M. Lamont et al.
on collisions can be best optimized by deploying round beams (equal beam sizes
and emittances in both planes). The generation of equal betatron functions at the IP
can be best provided by a triplet quadrupole configuration of the focusing elements
next to the IP (this implies equal β ∗ values in both planes for equal peak betatron
functions for both planes inside the triplet magnets). The beam-beam parameter in
this case is given by Eq. (9.74) and is entirely independent from the beta function
values at the IP. Optimizing the instantaneous luminosity in this case therefore
implies:
• increasing the beam brightness (N b // x,y ) until the beam-beam limit is reached;
• increasing the bunch population at constant brightness;
• minimizing the beta functions at the IP in both planes.
The second point is limited by the available aperture along the whole collider and by
collective effects that might limit the maximum bunch intensity. The third point is
limited by the available aperture of the triplet elements next to the IP (the maximum
beta function in these elements is proportional to 1/β ∗ ), the ‘hour glass’ effect
and eventual chromatic aberrations due to large betatron functions in the focusing
elements. The hour glass effect depends strongly on the bunch length and is only
relevant for a configuration with β ∗ ≤ σ s . Shortening the bunch length might be
another strategy for maximizing the luminosity in this case but the minimum bunch
length might itself be limited by other constrains such as the available RF voltage
and limitations on the maximum acceptable Intra-Beam-Scattering (IBS) growth
rates.
The maximum number of bunches in the collider is then either limited by
collective effects (e.g. electron-cloud effect for proton bunches), the maximum
acceptable stored beam power (e.g. machine protection issues) and the rate of
particle production in case the collider uses anti-protons in the collisions (e.g. the
Tevatron).
9.8.5 Luminosity Optimization in the Presence of a Crossing
Angle
If the operation of the collider permits a large number of bunches, it might be
necessary to introduce a crossing angle at the IP in order to avoid unwanted
collisions of the two beams next to the IP (e.g. the KEK-B factory and the LHC).
A crossing angle reduces the luminosity and the beam-beam parameter via the
geometric form factor F in Eq. (9.70). The geometric form factor is a function
of the crossing angle, the bunch length and the transverse beam size in the plane
of the crossing angle and therefore crossing angle operation introduces additional
parameter dependencies which make the luminosity optimization more complex.