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 = alphaJ(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 (e) in the ranges B less than or equal to 15 T, 4.2 K less than or equal to T less than or equal to 16 K and - 1.79% less than or equal to epsilon less than or equal to +0.67%. Complementary resistivity measurements were taken to determine the upper critical field (B-C2(T, epsilon)) and the critical temperature (T-C(epsilon)) directly. The upper critical field defined at 5%rho(N), 50%rho(N) or 95%rho(N), is described by the empirical relation B-C2(rhoN)(T, epsilon) = B-C2(rhoN)(0, epsilon) [1 - (T/T-C(rhoN)(epsilon))(nu)] The upper critical field at zero Kelvin and the critical temperature are linearly related where B-C2(rhoN)(0, epsilon) approximate to 3.6 T-C(rhoN)(epsilon) - 29.9, although strictly B-C2(rhoN) (0, epsilon) is a double-valued function of T-C(rhoN)(epsilon). J(E) was confirmed to be reversible at least in the range -0.23% < epsilon < 0.67%. The J(E) data have been parameterized using the volume pinning force (F-P) where F-P = J(E) x B = A(epsilon) B-C2(n)(T, epsilon)b(p)(1 - b)(q) and b = B/B-C2(T, epsilon). A(epsilon) is taken to be a C2 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 (similar to1 A), the data were divided into three temperature-strain ranges, B-C2(T, epsilon) described by the empirical relation and the constants p, q, n and v and the strain-dependent variables A(epsilon), B-C2(0, epsilon) and T-C(epsilon) 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, epsilon) using the resistivity data at 5% rho(N) where nu = 1.25, n = 2.18, p = 0.39 and q = 2.16. When B-C2 (T, epsilon) is constrained at 50% rho(N) or 95% rho(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%rhoN) (T,epsilon) 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 F-p = 1/249 [B-C2(T,epsilon)]5/2/(2piPhi(0))(1/2)mu(0)kappa(2)(T,epsilon)b(1/2)(1 - b)(2), where the Ginzburg-Landau (GL) parameter kappa is given by the relation kappa(T,epsilon) = 924B(C2)(T,epsilon)/gamma(1/2)(epsilon)T-C(epsilon)(1 - t(2)) gamma is the Sommerfeld constant and t = T/T-C(epsilon). At an applied field equal to the upper critical field found from fitting the Kramer dependence (i.e. at B-C2 (T, epsilon)), 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.