Geometrical alignment and associated beam optics issues of transfer lines with horizontal and vertical deflection

Skew quadrupoles are well known for their capability of tilting beams around their longitudinal axes. However, dipoles can produce the same effect. A horizontal bending dipole aligned with an inclination (slope) angle phi with respect to the reference coordinate system introduces a small tilt (roll) angle psi into the beam, which for small angles is proportional to the product of the magnet bending angle alpha and the slope phi. For transfer lines this effect has previously been neglected (it is comparable to the tilt alignment tolerance of the magnets of the order of 0.1 mrad rms), but for long lines with many horizontal bending magnets on relatively steep vertical slopes relative to the reference frame, a non-negligible tilt angle psi can accumulate between the local beam plane and the reference alignment system. This can result in mismatch at injection with subsequent emittance blow-up, and other undesirable effects, such as non-orthogonal trajectory measurement and correction. A detailed description of this effect is given, illustrated by the SPS to LHC transfer lines TI 8 and the LHC machine itself. The implications and the resulting choice of methodology for alignment of the elements along the line are described and a solution for matching to the plane of the LHC is outlined.