The strain and temperature scaling law for the critical current density of a jelly-roll Nb₃Al strand in high magnetic fields

The engineering critical current density (J_E) and the index of transition, N (where E = αJ^N), of a Nb₃Al multifilamentary strand, mass-produced as a part of the Fusion programme, have been characterized as a function of field (B), temperature (T) and strain (ε) in the ranges B ≤ 15 T, 4.2 K ≤ T ≤ 16 K and −1.79% ≤ ε ≤ +0.67%. Complementary resistivity measurements were taken to determine the upper critical field (B_C2(T, ε)) and the critical temperature (T_C(ε)) directly. The upper critical field defined at 5%ρ_N, 50%ρ_N or 95%ρ_N, is described by the empirical relation B_C2^ρ_N(T, ε) = B_C2^ρ_N(0, ε)[1 −(T/T_C^ρ_N(ε))^ν]. The upper critical field at zero Kelvin and the critical temperature are linearly related where B_C2^ρ_N (0, ε) ≈ 3.6T_C^ρ_N (ε) − 29.9, although strictly B_C2^ρ_N (0, ε) is a double-valued function of T_C^ρ_N (ε). J_E was confirmed to be reversible at least in the range −0.23% < ε < 0.67%. The J_E data have been parameterized using the volume pinning force (F_P) where F_P = J_E × B = A(ε)B_C2ⁿ (T, ε)b^p (1 − b)^q and b = B/B_C2(T, ε). A(ε) is taken to be a function of strain otherwise the maximum value of F_P (found by varying the field) was a double-valued function of B_C2 when the temperature was fixed and the strain varied. To achieve a very high accuracy for the parameterization required by magnet engineers (∼1 A), the data were divided into three temperature–strain ranges, B_C2(T, ε) described by the empirical relation and the constants p, q, n and ν and the strain-dependent variables A(ε), B_C2(0, ε) and T_C(ε) treated as free-parameters and determined in each range. A single scaling law that describes most of the J_E data has also been found by constraining B_C2(T, ε) using the resistivity data at 5%ρ_N where ν = 1.25, n = 2.18, p = 0.39 and q = 2.16. When B_C2(T, ε) is constrained at 50%ρ_N or 95%ρ_N, the scaling law breaks down such that p and q are strong functions of temperature and q is also a strong function of strain. Good scaling provides support for identifying B_C2^5%ρ_N (T, ε) as the characteristic (or average) upper critical field of the bulk material. The J_E data are also consistent with a scaling law that incorporates fundamental constants alone, of the Kramer-like form where the Ginzburg–Landau (GL) parameter κ is given by the relation γ is the Sommerfeld constant and t = T/T_C(ε). At an applied field equal to the upper critical field found from fitting the Kramer dependence (i.e. at B_C2(T, ε)), the critical current is non-zero and we suggest that the current flow is percolative. The functional form of F_P implies that in high fields the grain boundary pinning does not limit J_E, this is consistent with J_E-microstructure correlations in other superconducting materials.