4 Impedance and Collective Effects
161
4.6.7.3 Strength of Long-Range Interactions
Assuming a separation d in the horizontal plane, the kicks in the two planes can be
written as
Δx
= −
2Nr 0
γ
(x + d)
r 2
1 − e
−
r 2
2σ 2
(4.65)
with r 2 = (x + d) 2 + y 2 . The equivalent formula for the plane orthogonal to the
separation is
Δy
= −
2Nr 0
γ
y
r 2
1 − e
−
r 2
2σ 2
(4.66)
The effect of long-range interactions must strongly depend on the separation.
The calculation shows that the tune spread ΔQ lr from long-range interactions alone
follows an approximate scaling (for large enough separation, i.e. above ~6σ )
ΔQ lr ∝ −
N
d 2
(4.67)
where N is the bunch intensity and d the separation. Small changes in the separation
can therefore result in significant differences. Since the symmetry between the two
planes is broken, the resulting footprint shows no symmetry. In fact, the tune shifts
have different signs for x and y, as expected.
4.6.7.4 Footprint for Long-Range Interactions
Contrary to the head-on interaction where the small amplitude particles are mostly
affected, now the large amplitude particles experience the strongest long-range
beam–beam perturbations. This is rather intuitive since the large amplitude particles
are the ones which can come closest to the opposing beam as they perform their
oscillations. We must therefore expect a totally different tune footprint. Such a
footprint for only long-range interactions is shown in Fig. 4.26.
4.6.8 Studies of Long Range Interactions in the LHC
To study the effect of long range beam–beam interactions we have performed a
dedicated experiment [172]. The LHC was set up with single trains of 36 bunches
per beam, spaced by 50 ns. The bunch intensities were ~1.2 × 10 11 p/b and the
normalized emittances around 2.5 μm. The trains collided in IP1 and IP5, leading to
a maximum of 16 long range encounters per interaction point for nominal bunches.
First, the crossing angle (vertical plane) in IP1 was decreased in small steps and the
161
4.6.7.3 Strength of Long-Range Interactions
Assuming a separation d in the horizontal plane, the kicks in the two planes can be
written as
Δx
= −
2Nr 0
γ
(x + d)
r 2
1 − e
−
r 2
2σ 2
(4.65)
with r 2 = (x + d) 2 + y 2 . The equivalent formula for the plane orthogonal to the
separation is
Δy
= −
2Nr 0
γ
y
r 2
1 − e
−
r 2
2σ 2
(4.66)
The effect of long-range interactions must strongly depend on the separation.
The calculation shows that the tune spread ΔQ lr from long-range interactions alone
follows an approximate scaling (for large enough separation, i.e. above ~6σ )
ΔQ lr ∝ −
N
d 2
(4.67)
where N is the bunch intensity and d the separation. Small changes in the separation
can therefore result in significant differences. Since the symmetry between the two
planes is broken, the resulting footprint shows no symmetry. In fact, the tune shifts
have different signs for x and y, as expected.
4.6.7.4 Footprint for Long-Range Interactions
Contrary to the head-on interaction where the small amplitude particles are mostly
affected, now the large amplitude particles experience the strongest long-range
beam–beam perturbations. This is rather intuitive since the large amplitude particles
are the ones which can come closest to the opposing beam as they perform their
oscillations. We must therefore expect a totally different tune footprint. Such a
footprint for only long-range interactions is shown in Fig. 4.26.
4.6.8 Studies of Long Range Interactions in the LHC
To study the effect of long range beam–beam interactions we have performed a
dedicated experiment [172]. The LHC was set up with single trains of 36 bunches
per beam, spaced by 50 ns. The bunch intensities were ~1.2 × 10 11 p/b and the
normalized emittances around 2.5 μm. The trains collided in IP1 and IP5, leading to
a maximum of 16 long range encounters per interaction point for nominal bunches.
First, the crossing angle (vertical plane) in IP1 was decreased in small steps and the