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

Abstract

The engineering critical current density (JE) and the index of transition, N (where E = αJN), 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 (BC2(T, ε)) and the critical temperature (TC(ε)) directly. The upper critical field defined at 5%ρN, 50%ρN or 95%ρN, is described by the empirical relation BC2ρN(T, ε) = BC2ρN(0, ε)[1 −(T/TCρN(ε))ν]. The upper critical field at zero Kelvin and the critical temperature are linearly related where BC2ρN (0, ε) ≈ 3.6TCρN (ε) − 29.9, although strictly BC2ρN (0, ε) is a double-valued function of TCρN (ε). JE was confirmed to be reversible at least in the range −0.23% < ε < 0.67%. The JE data have been parameterized using the volume pinning force (FP) where FP = JE × B = A(ε)BC2n (T, ε)bp (1 − b)q and b = B/BC2(T, ε). A(ε) is taken to be a function of strain otherwise the maximum value of FP (found by varying the field) was a double-valued function of BC2 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, BC2(T, ε) described by the empirical relation and the constants p, q, n and ν and the strain-dependent variables A(ε), BC2(0, ε) and TC(ε) treated as free-parameters and determined in each range. A single scaling law that describes most of the JE data has also been found by constraining BC2(T, ε) using the resistivity data at 5%ρN where ν = 1.25, n = 2.18, p = 0.39 and q = 2.16. When BC2(T, ε) is constrained at 50%ρN or 95%ρN, 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 BC25%ρN (T, ε) as the characteristic (or average) upper critical field of the bulk material. The JE data are also consistent with a scaling law that incorporates fundamental constants alone, of the Kramer-like formwhere the Ginzburg–Landau (GL) parameter κ is given by the relationγ is the Sommerfeld constant and t = T/TC(ε). At an applied field equal to the upper critical field found from fitting the Kramer dependence (i.e. at BC2(T, ε)), the critical current is non-zero and we suggest that the current flow is percolative. The functional form of FP implies that in high fields the grain boundary pinning does not limit JE, this is consistent with JE-microstructure correlations in other superconducting materials.