Unified Scaling Law for flux pinning in practical superconductors: II. Parameter testing, scaling constants, and the Extrapolative Scaling Expression

Abstract A scaling study of several thousand Nb 3 Sn critical-current ( I c ) measurements is used to derive the Extrapolative Scaling Expression (ESE), a relation that can quickly and accurately extrapolate limited datasets to obtain full three-dimensional dependences of I c on magnetic field ( B ), temperature ( T ), and mechanical strain ( ε ). The relation has the advantage of being easy to implement, and offers significant savings in sample characterization time and a useful tool for magnet design. Thorough data-based analysis of the general parameterization of the Unified Scaling Law (USL) shows the existence of three universal scaling constants for practical Nb 3 Sn conductors. The study also identifies the scaling parameters that are conductor specific and need to be fitted to each conductor. This investigation includes two new, rare, and very large I c ( B , T , ε ) datasets (each with nearly a thousand I c measurements spanning magnetic fields from 1 to 16 T, temperatures from ∼2.26 to 14 K, and intrinsic strains from –1.1% to +0.3%). The results are summarized in terms of the general USL parameters given in table 3 of Part 1 (Ekin J W 2010 Supercond. Sci. Technol. 23 083001 ) of this series of articles. The scaling constants determined for practical Nb 3 Sn conductors are: the upper-critical-field temperature parameter v = 1.50 ± 0.04; the cross-link parameter w = 3.0 ± 0.3; and the strain curvature parameter u = 1.7 ± 0.1 (from equation (29) for b c2 ( ε ) in Part 1). These constants and required fitting parameters result in the ESE relation, given by I c ( B,T,ε ) B = C [ b c2 ( ε )] S (1− t 1.5 ) η − μ (1- t 2 ) μ b p (1− b ) q with reduced magnetic field b ≡ B / B c2 *( T , ε ) and reduced temperature t ≡ T / T c *( ε ), where: B c2 * ( T , ε ) = B c2 * (0,0)(1− t 1.5 ) b c2 ( ε ) T c * (ε)= T c * (0)[ b c2 ( ε )] 1/3 and fitting parameters: C , B c2 *(0,0), T c *(0), s , either η or μ (but not both), plus the parameters in the strain function b c2 ( ε ). The pinning-force shape parameters p and q are also preferably fitted (simultaneously with the other parameters), but default values p = 0.5 and q = 2.0 also give high fitting accuracy when the range of relative magnetic fields is not extensive. Default values are also essential when the magnetic field data range is insufficient to determine p and q . The scaling constants are remarkably stable (changes less than ∼1%) with respect to different values of p and q , Nb 3 Sn conductor configurations, magnetic self-field corrections, and pinning-force trim values. The results demonstrate that the scaling of transport critical current holds down to the lowest temperatures measured ∼2.2 K, for both magnetic self-field corrected and uncorrected data. An initial comparison is also made between transport and magnetization scaling data in matched Nb 3 Sn samples and significant differences are found, especially for the upper critical field B c2