Coil End Optimization of the Nb$_3$Sn Quadrupole for the High Luminosity LHC

As part of the Large Hadron Collider Luminosity upgrade (HiLumi-LHC) program, the US LARP collaboration and CERN are working together to design and build a 150 mm aperture quadrupole magnet that aims at providing a nominal gradient of 140 T/m. The resulting conductor peak field of more than 12 T requires the use of Nb$_{3}$Sn superconducting coils. In this paper the coil design for the quadrupole short model (SQXF) is described, focusing in particular on the optimization of the end-parts. We first describe the magnetic optimization aiming at reducing the peak field enhancement in the ends and minimizing the integrated multipole content. Then we focus on the analysis and tests performed to determine the most suitable shapes of end turns and spacers, minimizing the mechanical stress on the cables. We conclude with a detailed description of the baseline end design for the first series of the short model coils.