Magnet sorting is a challenging process that aims at improving the performance of a particle accelerator by the clever choice of the position of the magnets to induce a natural compensation of the field errors that are inevitable in modern superconducting magnets.
These studies are intrinsically multi-faceted and an appropriate figure of merit should be found that efficiently characterises the accelerator performance. This aspect not only involves beam dynamics concepts, but also entails computational considerations, as magnet sorting studies face the explosion of computational time due to the intrinsic combinatorial character of the problem. Furthermore, an efficient sorting relies on the availability of a sufficiently large pool of magnets so that installation slots can be optimised. This is an additional challenge, as it tends to conflict with the optimisation of the production and installation schedule, imposing an external constraint on magnet production. Finally, magnet sorting requires a detailed knowledge of the magnet properties, which requires a timely and detailed programme of magnetic measurements, ideally at nominal operating temperatures. All these aspects should be perfectly synchronised and harmonised to provide the best conditions to exploit the benefits of magnet sorting.
In the case of the HL-LHC upgrade, magnet sorting is envisaged for the triplet quadrupoles. These are produced by CERN and the Accelerator Upgrade Program (AUP), necessitating a perfect coordination between the involved international collaborators.
Despite the efforts to make the final cryo-objects as similar as possible, there are several special hardware features that are linked to the final position in the tunnel. The decision on slot allocation, which is the outcome of the sorting studies, should therefore be taken early on to maximise the number of slots compatible with each cryo-object, however to take such a decision requires the results of detailed magnetic measurements, which increases the time pressure on the whole process.
What is already clear is that, from the beam dynamics point of view, the key observable is the transfer function of the quadrupole magnets. Deviations from the nominal value generate perturbations of the accelerator optics - the so-called beta-beating - which impact the overall performance by altering the luminosity value and its balance between the two high-luminosity experiments. Furthermore, certain configurations of beta-beating might be hard to compensate based on beam-based measurements and corrector magnets, so this should be avoided at all costs during sorting.
Beta-beating affects both horizontal and vertical motions so that the most efficient score function consists of the sum in quadrature of the RMS (computed over the accelerator circumference) beta-beating in the horizontal and vertical planes. Numerical simulations, including the analysis of all possible combinations of positions for Q1 and Q3 magnets, or Q2A and Q2B magnets, confirm the sizeable reduction of the score function thanks to an appropriate sequence of the quadrupole magnets (see figure 1).
The next steps will include the consideration of more realistic constraints from the standpoint of hardware and the use of measured values of the transfer function. On the hardware side, options to pair magnets in Q1 or Q3 cryostats are being considered, which might also be used to improve the field quality of the pair of magnets in a given cryostat.
