Unified scaling law for flux pinning in practical superconductors: I. Separability postulate, raw scaling data and parameterization at moderate strains

The unified strain-and-temperature scaling law underlies the many pinning-force-model expressions proposed to parameterize the dependence of the critical current Ic on magnetic field B, temperature T and applied axial strain ε in superconductors for high-field magnet design. Increasingly, these expressions have been evaluated by use of multiparameter simultaneous fits, without the use of scaling. In this review, we reintroduce the unified scaling law on which the recent parameterizations are based, as well as the power of raw scaling data for parameter consistency, extrapolation capability and data-based model evaluation. The unified scaling law (USL) for the flux-pinning force per unit conductor length in practical high-field superconductors is expressed by where K(t, ε0) is a temperature-and-strain dependent prefactor. The scaling variables are: reduced magnetic field with Bc2*(t, ε0) an effective upper critical field; reduced temperature with Tc*(ε0) an effective strain-dependent critical temperature; and intrinsic axial strain defined as zero at the strain εm, where Ic is maximum. The scaling parameters p and q are constants, which is a necessary (but not sufficient) condition for scaling. The strain dependences of Tc* and Bc2* are correlated as where and (bc2(ε0) is also designated as s(ε0) in the recent literature). It is shown that, when raw scaling data are used to determine the scaling parameter w, it has a constant value w≈3.0 ± 0.1 in a wide range of Nb3Sn superconductors. This is significant, because w is the only scaling parameter that requires very large data matrices of Ic(B, T, ε) to determine its value. Unified scaling has been demonstrated in a number of different superconducting materials, including Nb3Sn, Nb3Al and more recently, the fundamentally distinct MgB2 and Bi-2223 material systems. The separable form of the USL is a great simplification in which the USL is parameterized in terms of separate functions of intrinsic strain ε0, reduced temperature t and reduced magnetic field b, given by with the following dimensionless notation: , and . C is a proportionality constant. With this separable form, the scaling law is broken down into five dimensionless scaling functions with values ranging from 0 and 1: bc2(ε0), bc2(t), g(ε0), h(t) and f(b). Each of these separate functions depends individually on strain, reduced temperature or reduced magnetic field, with no commingled variables. We show that most parameterizations recently proposed for the USL can be broken down into these five single-variable scaling functions, providing the ability to determine the free scaling parameters in small groups (<4 free variables each), which greatly improves their fitting accuracy and consistency. A table comparing USL parameterizations in terms of these separable functions serves as an effective ‘Rosetta Stone’ table for translating among the various parameterizations. An accurate, simple parameterization of the USL is given in separable form for Nb3Sn for the less complex and less controversial moderate strain regime (−0.5% < ε0 < ε0irr), where many magnets are designed: with free scaling parameters as follows: strain parameters εm, s, a − (for ε0 < 0) and a + (for ε0 > 0); temperature parameters Tc*(0) and η; magnetic field parameters Bc2*(0, 0), p and q; and C is a proportionality constant. With this parameterization, the scaling parameters themselves are also separable, an important feature for practical engineering purposes, because the parameter values can be built up from separate strain and temperature measurements. The only non-separable parameter, w, is fixed at 3.0, as described above. The parameter μ is fixed at 0, 1 or 2, corresponding to the three parameterization models in present use for the temperature function h(t), all of which are effectively equivalent in fitting accuracy at T ≥ 4 K (the simplest being the original parameterization μ = 0). At high compressive strains (ε0 < − 0.5%), a consensus for the best parameterizations has not yet been achieved. In Part II of this review, raw scaling data will be used to assess the most commonly used parameterizations in this regime, especially bc2(ε0) and g(ε0) at high compressive strains, and h(t) over a wide temperature range.