Correlation Between the Number of Displacements Per Atom and $T_c$ After High-Energy Irradiations of Nb$_3$Sn Wires for the HL-LHC

In order to predict the irradiation effects in the quadrupoles of the High-Luminosity Large Hadron Collider accelerator at the European Organization for Nuclear Research (CERN) during operation up to a luminosity of 4000 fb^-1, an irradiation program was carried out on industrial Taand Ti-added Nb$_{3}$Sn wires. Wire samples from the same batch were irradiated with high-energy protons (65 MeV and 24 GeV, up to 1.4×10^21 m^-2) and neutrons (> 0.1 MeV, up to 1.8×10^22 m^-2). The values of T' and Jc were reported as a function of particle fluence. After replacing the fluence by the number of displacements per atom (dpa), which is determined using the FLUKA code, it was found that the variation of T_c in Nb_3Sn wires as a function of the dpa value for both proton and neutron irradiation falls on the same curve, reflecting a universal behavior. This result reflects the fact that the variation of T_c is uniquely governed by the change in atomic ordering S. Both the measured value T_c and the calculated one dpa essentially depend on the number of Frenkel defects. With the new relationship between T_c and dpa for both protons and neutrons, the decrease of T_c in the quadrupoles at the maximum luminosity can be estimated to ~0.3 K. The variation of J_c versus dpa shows some similarities between proton and neutron irradiation, too, but the analysis is more complex, the observed enhancement of J_c with irradiation being due to enhanced point pinning caused by the radiation-induced defect clusters.