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

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 $B_{c2}*(T,ε)$, which may be a result of inhomogeneous shielding currents. In Part 3 of this topical review series (Ekin J W 2017 Supercond. Sci. Technol. at press), the smallest practical minimum dataset for extrapolating full I c(B,T,ε) datasets is derived. Application of the ESE relation is illustrated in several new areas, including full characterization of Nb$_{3}$Sn conductors from as little as a single $I_c(B)$ curve when a few core parameters have been determined for similar conductors.