Correction of vertical crossing induced dispersion in LHC Page: 4 of 16
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1 Introduction
Crossing angle and orbit off-centering schemes at interaction points (IP) in LHC ring in collision
optics mode are foreseen [1, 2], for full separation of the beams during energy ramping phase, or
early separation of the beams beyond the IP during collision, in order to reduce as much as possible
harmful effects related to beam-beam interactions in that region where they share a common vacuum
pipe. Both planes may be affected by crossing or off-centering, e.g., in the 45 deg. inclined crossing
plane scheme. In terms of orbit design this means non-zero closed orbit (c.o.) angle (crossing) or
non-zero c.o. off-centering (separation) at the IP of concern, and in consequence in the low-4 triplets,
which has a sensible effect on the dispersion function in collision optics when betatron functions reach
very large values. In terms of the equations of motion, the non-zero c.o. induces dispersive terms of
first order in momentum deviation, with corresponding particular closed solution.
This phenomenon has been subject to detailed investigation in Ref. [3] in the frame of the
LHC Version 4.2. We now address the recently designed Version 5.0 [5] of the ring. The main
aspects of dispersion excitation are recalled ; it is shown that in the nominal optical conditions
(Q,/Qy = 63.32/59.31, 0.1 mrad vertical crossing, * = 0.5 m) and under propitious betatron
phase relations between IP's, the so induced vertical dispersion may reach the limit of tolerances
in the case of a single crossing and even exceed it in the case of several crossings. It is also shown
that prohibitively large figures are attained in the extreme 4-squeeze conditions ( 0.2 to 0.4 mrad
vertical crossing, 4* = 0.25 m), therefore justifying foreseeing a local correction scheme.
Correction strategies for horizontal crossing induced dispersion have already been investigated in
detail and are now part of the LHC design [4]. Vertical crossing induced dispersion and correction
principles for its compensation have also been investigated in Ref. [3], however a practical correction
scheme for LHC V5.0 still remained to be defined, which is done here. The device is based on the
use of skew quadrupoles located as close as possible to the low-4 triplets at the neighbouring arc
ends. Corrector strengths are derived analytically and allow quantifying the needs for Version 5.0.
The report is organized as follows. In Section 2 the differential equation for the vertical crossing
induced dispersion is established and its effects are derived and quantified. Section 3 describes
the proposed correction optics. Numerical applications and simulations undertaken in the report
are based on the Version 5.0 of the LHC optics [5]. MAD [6] simulations are performed wherever
necessary, with the regular LHC lattice files [7] and preliminary 4* = 0.25 m optics [9]. The present
work largely leans on Ref. [3] which may in particular be referred to for comparison with prior similar
study involving Version 4.2 of the optics.
2 Vertical crossing/off-centering induced dispersion
2.1 Perturbative periodic dispersion ; scaling
Vertical dispersive effects related to c.o. geometry derive from the equation of motion
d2yr/ds2 + K(s)yr = -(1 - 6)ABy(s)/Bp + K(s)yrd (1)
in which yr is the transverse excursion w.r.t. machine axis, Bp is the particle rigidity, K(s) the
quadrupole strength and S the momentum deviation. The field term -ABy(s)/Bp is due to the
c.o. dipoles and its factor (1 - 3) accounts for their first order chromatic effect. The second order
dispersive term K(s)yrd is due to quadrupoles. Taking yr = y + yco (Yco = c.o. excursion, y =
particle excursion w.r.t. c.o.) leads to the differential equation
d2Dy/ds2 + K(s)Dy ABy(s)/Bp + K(s)yco (2)
for the vertical dispersion Dy = y/6. The elementary kick approximation K(s)yco (s) = f K(s)yco (s)(s-
sq)dsq [S(s - sq) = Dirac impulse at azimuth sq], yields the solution2
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Meot, F. Correction of vertical crossing induced dispersion in LHC, report, November 1, 1997; Batavia, Illinois. (https://digital.library.unt.edu/ark:/67531/metadc695811/m1/4/: accessed April 22, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.