Extrapolative Scaling Expression: A Fitting Equation for Extrapolating Full $I_c (B,T,ϵ)$ Data Matrixes From Limited Data

Scaling analysis of several thousand $Nb_3Sn$ critical-current $(I_c)$ measurements is used to derive the extrapolative scaling expression (ESE), a fitting equation that can quickly and accurately extrapolate (or interpolate) limited datasets to obtain full three-dimensional dependences of $I_c$ on magnetic field (B), temperature (T), and mechanical strain (ε). Unlike nonextrapolative fitting equations, the ESE relation is determined completely by fundamental raw scaling data from master pinning-force curves. The results show that extrapolation errors with ESE approach typical $I_c$ measurement errors. The scaling expression is simple and robust, providing straightforward extrapolation capability for conductor characterization and magnet design. The ESE relation offers the prospect for extrapolations in several new areas, including: a reduction in the measurement space required for full $I_c(B, T, ε)$ characterization to about one fifth the size; ability to combine data from separate temperature and strain apparatuses; extrapolation of transport $I_c$ data from above 4 K to lower temperatures, where heating effects and instabilities are problematic for transport measurements; and extrapolation of full $I_c(B, T, ε)$ datasets from as little as a single $I_c(B)$ curve when several core parameters have been measured in similar conductors (particularly applicable to qualifying production wires). Accuracies are evaluated for concatenations of these different extrapolation capabilities. Examples are given for practical $Nb_3Sn$ conductors, including those for high luminosity-LHC magnets, ITER, and cryocooled NMR magnets.