A general scaling relation for the critical current density in Nb3Sn

We review the scaling relations for the critical current density (J(c)) in Nb3Sn wires and include recent findings on the variation of the upper critical field (H-c2) with temperature (T) and A15 composition. Measurements of H-c2(T) in inevitably inhomogeneous wires, as well as analysis of literature results, have shown that all available H-c2(T) data can be accurately described by a single relation from the microscopic theory. This relation also holds for inhomogeneity averaged, effective, H-c2(*)(T) results and can be approximated by H-c2(t)/H-c2(0) congruent to 1 - t(1.52), with t = T/T-c. Knowing H-c2(*)(T) implies that J(c)(T) is also known. We highlight deficiencies in the Summers/Ekin relations, which are not able to account for the correct J(c)(T) dependence. Available J(c)(H) results indicate that the magnetic field dependence for all wires from mu H-0 congruent to 1 T up to about 80% of the maximum H-c2 can be described with Kramer's flux shear model, if nonlinearities in Kramer plots when approaching the maximum H-c2 are attributed to A15 inhomogeneities. The strain (epsilon) dependence is introduced through a temperature and strain dependent H-c2*(T, epsilon) and Ginzburg-Landau (GL) parameter kappa(1)(T, epsilon) and a strain dependent critical temperature T-c(epsilon). This is more consistent than the usual Ekin unification of strain and temperature dependence, which uses two separate and different dependences on H-c2(*)(T) and H-c2(*)(epsilon). Using a correct temperature dependence and accounting for the A15 inhomogeneities leads to the remarkably simple relation J(c)(H, T, epsilon) congruent to (C/mu H-0)s(epsilon)(1 - t(1.52)) (1 - t(2))h(0.5)(1 - h)(2), where C is a constant, s(epsilon) represents the normalized strain dependence of H-c2*(0) and h = H/ H-c2*(T, epsilon). Finally, a new relation for s(epsilon) is proposed, which is an asymmetric version of our earlier deviatoric strain model and based on the first, second and third strain invariants. The new scaling relation solves a number of much debated issues with respect to J(c) scaling in Nb3Sn and is therefore of importance to the applied community, who use scaling relations to analyse magnet performance from wire results.