Study of the Stability of the LHC Low Beta Inner Triplet for a NB₃SN design

The inner triplet of the low β insertion of LHC is made up of four identical NbTi superconducting quadrupoles. To achieve the nominal luminosity of 10³4cm^-2s^-1 a gradient of 235 T/m is necessary with an aperture of 70 mm. Such a gradient yields a peak field on the coils close to the critical field, endangering the stability for NbTi windings. Improving in the beam and/or collision dynamics depends on the focussing strength of the quads. Nb₃Sn magnet, technologically more complex of the NbTi ones, are a promising solution to improve the gradient beyond the 250 T/m; for this reason we undertook the study of a second generation of low β insertions with N₃Sn magnets. Critic for both the construction techniques is the power deposed by the secondary particles generated by the collisions of the 7+7 TeV proton beams. Previous studies estimated the total power released in the NbTi coils to be around 30 W. In this paper different configurations of aperture and gradient of Nb3Sn coils are examined by tracking the reaction products, as from DTUJET code, in the magnetic field structure and evaluating the energy deposition with the FLUKA code. The stability margin of the magnets is then stated with the analysis of the thermal behaviour with the ANSYS code. The better thermal behaviour using aluminum collars instead of stainless steel is shown. The use of cylindrical absorbers under the magnet leads to an unexpected increase of the peak and of the total power in the second quad of about 50% and of 20% respectively. The operating conditions of all the configurations guarantee a safety margin of operation with no need of any kind of absorbers. In the reference case the maximum power and the peak power deposed in the second quadrupole is 22. 4 W and 7. 2 mW/cm³ respectively. The study shows Nb₃Sn seems a viable solution for the low b insertion for LHC if gradient in excess of 250 T/m will be required.