Weakly-Emergent Strain-Dependent Properties of High Field Superconductors

All superconductors in high field magnets operating above 12 T are brittle and subjected to large strains because of the differential thermal contraction between component parts on cool-down and the large Lorentz forces produced in operation. The continuous scientific requirement for higher magnetic fields in superconducting energy-efficient magnets means we must understand and control the high sensitivity of critical current density Jc to strain ε. Here we present very detailed Jc(B, θ, T, ε) measurements on a high temperature superconductor (HTS), a (Rare−Earth)Ba2Cu3O7−δ (REBCO) coated conductor, and a low temperature superconductor (LTS), a Nb3Sn wire, that include the very widely observed inverted parabolic strain dependence for Jc(ε). The canonical explanation for the parabolic strain dependence of Jc in LTS wires attributes it to an angular average of an underlying intrinsic parabolic single crystal response. It assigns optimal superconducting critical parameters to the unstrained state which implies that Jc(ε) should reach its peak value at a single strain (ε = εpeak), independent of field B, and temperature T. However, consistent with a new analysis, the high field measurements reported here provide a clear signature for weakly-emergent behaviour, namely εpeak is markedly B, (field angle θ for the HTS) and T dependent in both materials. The strain dependence of Jc in these materials is termed weakly-emergent because it is not qualitatively similar to the strain dependence of Jc of any of their underlying component parts, but is amenable to calculation. We conclude that Jc(ε) is an emergent property in both REBCO and Nb3Sn conductors and that for the LTS Nb3Sn conductor, the emergent behaviour is not consistent with the long-standing canonical explanation for Jc(ε).

The critical current density J c is the maximum current density that can be carried by a superconductor before significant dissipation results from flux flow. It is the most important parameter in high field magnet design for systems such as MRI 1 , particle accelerators 2 and fusion energy reactors 3 . In high field superconductors, J c is usually parameterised in terms of a volume flux pinning force F p and is a function of magnetic field B, temperature T, applied uniaxial strain ε app and for an anisotropic conductor, the angle θ between B and say the normal to the tape surface.
There are various theories of flux pinning that describe J c . Theory 4,5 and experiment 6,7 often lead to the same generalised scaling law of the form where B c2 is the upper critical field, κ 1 is the Ginzburg-Landau parameter, b = B/B c2 is the reduced field, μ 0 is the vacuum permeability, φ 0 is the magnetic flux quantum, A is a material dependent constant, and n, m, p and q are constants dependent on the specific pinning mechanism operating. The flux pinning scaling law is widely observed in many different types of superconducting materials including low temperature superconductors (LTS) and high temperature superconductors (HTS). This is because its form is derived using Ginzburg-Landau theory, which is founded on Landau's very general theory of second-order phase transitions, and can equally well describe superconductors with different microscopic fundamental mechanisms causing the superconductivity 8 .
coincidence between the Fermi energy and a peak in the density of states produced by the narrow d-band electrons in the Nb-chains 34 . In principle, this explains the relatively high values of T c , and the optimum values of J c occuring in the unstrained or zero intrinsic strain state (i.e. ε int = 0) 35 . The strain dependency of T c is attributed to variations in both phononic and electronic properties. In this canonical description the parameter ε peak specifies the optimum strain state, or equivalently the optimum atomic spacings in the material, for peak superconducting critical parameters such as T c , and therefore should not depend on B and T. Given the very good scaling of F p , it has also been assumed since then that all the material responds to an applied strain in a similar manner and hence measurements of J c provided averaged properties 6,7,15 . However even now, although Nb 3 Sn is to be used in the multi-billion dollar ITER fusion tokamak 36 and the LHC high-luminosity upgrade 37 , uniaxial strain dependent single crystal data (for Nb 3 Sn 38 or any A15 material 39 ) remain very limited. We have found that even in the limited data available, there is no experimental evidence for the optimum superconducting critical properties in single crystals occurring in the unstrained state. This undermines the generally accepted interpretation of ε peak that includes equating the strain dependent properties of polycrystalline Nb 3 Sn wires, such as T c and B c2 , to an angular average of single crystal properties 14 . We propose that although ε peak is the optimum strain for the overall properties of the material, one has to abandon the standard interpretation that ε peak is associated with the optimum properties for the component parts of the material. High J c wires of the type presented here are designed for high field operation. This makes them prone to instability in low fields and in practice has prevented any reports of experimental data describing the strain dependence of high J c Nb 3 Sn in zero-field. Here we present high field measurements and find, strikingly, that as with REBCO, ε peak is a marked function of B and T. Hence we conclude that J c (ε) in both REBCO and Nb 3 Sn is emergent.

Methods
Transport J c and B c2 measurements were performed on a HTS REBCO coated conductor manufactured by SuperPower 40 (Ref: SCS4050) using the four-probe method with a custom-built probe in our in-house 15 T liquid helium cooled, 40 mm wet-bore, superconducting, split-pair horizontal magnet 41 . The sample was soldered to the top of a springboard made of CuBe as shown in Fig. 1. Compressive and tensile strain can be applied to the sample by pulling apart or pushing together the legs of the springboard. Force was applied to the legs of the springboard using a pushrod attached to a screw jack with a high gearing ratio. The strain was monitored continuously using a strain gauge attached to the springboard alongside the sample next to the voltage taps. The voltage tap separation was 13 mm, located about the centre of the springboard. Temperature control was achieved through use of an inverted temperature cup 42 . The cup is sealed at the top and has a vent at the bottom as shown in Fig. 1. Initially it fills with liquid helium. Three heaters attached to the underside of the springboard drive the liquid helium out through the vent leaving a gaseous environment. The temperature of the sample was controlled by a temperature controller using the three heaters in conjunction with three field calibrated Cernox TM resistance thermometers attached to the top, the middle and the bottom of the sample. The field calibration for the thermometry was taken from literature 43 and confirmed in liquid helium at 4.2 K. J c measurements were performed holding the field, temperature and strain constant, and ramping the current at a rate such that each measurement took ~60 s. The voltage, current and temperature were measured continuously. A nanovolt amplifier with a gain of 50,000 was used to amplify the voltage signal and the current was determined by measuring the voltage drop across a calibrated low resistance shunt connected in series with the power supply and sample. The experimental setup is shown in Fig. 2. The current through the superconductor I SC is slightly lower than that supplied by the power supply I total due to current shunting through the sample holder and stabilising materials in the conductor. This was accounted for by subtracting the shunt current from the measured current using the equation SC total shunt where V is the measured voltage across the sample, and R shunt is the resistance of the sample holder and stabilising materials which was determined as a function of field and strain from the B c2 traces. The typical magnitude of the shunt current was 80 mA at 100 μV m −1 . The critical current was converted to a critical current density using the cross-sectional area of the superconductor, taken to be 4 × 10 −3 mm 2 . J c was determined at a critical E-field criterion of 100 μVm −1 , and the index of transition N by fitting the relation E ∝ J N between 10 and 100 μV m −1 . B c2 measurements were performed holding the field and strain constant. A small current of 100 mA was applied and the temperature increased to above the transition at a rate of 1 Kmin −1 . The voltage and temperature were measured continuously and B c2 was determined at the onset of the superconducting transition (i.e. close to 100 % of the normal state resistance of the stabilising matrix of the composite).
The sample was aligned with respect to the magnetic field using a Hall probe attached to the sample such that θ = 0° when the magnetic field was normal to the surface of the tape. Measurements were taken first at θ = 0°. The strain was taken to ε app = −1 % and held constant as J c and B c2 were obtained as a function of field and temperature. At temperatures of 4.2, 20, 40 and 60 K measurements of J c were taken from 2 to 14 T in intervals of 2 T or until I total > 250 A (the maximum current the probe can sustain). At temperatures of 68 and 76 K measurements were taken at 1 T intervals up to 14 T or until B > B c2 . B c2 measurements were taken at fields of 0 to 14 T in intervals of 2 T. The strain was then increased in intervals of 0.25 % to +0.5 % and held constant at each strain where another field and temperature dependent dataset was obtained. To ensure the sample was undamaged by the strain cycle, eventually the applied strain was relaxed to zero and measurements of J c at 2 T and 60 K, and B c2 at 2 T were taken and were found to agree with the results taken at the start of the experiment.
Dense J c measurements were then taken as a function of angle to complement the data taken at fixed angle. The peak in J c , when the field is aligned with the ab-plane, was found at θ = 87.5° showing there was a −2.5° (2019) 9:13998 | https://doi.org/10.1038/s41598-019-50266-1 www.nature.com/scientificreports www.nature.com/scientificreports/ difference between the ab-plane and the tape surface. The dense angular measurements were used to select four angles at which to perform detailed strain dependent measurements θ = 47.5°, 77.5°, 82.5° and 87.5° which cover a large range in J c . The strain was taken to ε app = −1 % and held as J c measurements were obtained as a function of angle, at temperatures of 20, 40 and 60 K and fields from 2 to 14 T in intervals of 2 T or until I total > 250 A. The strain was then increased in intervals of 0.25 % to +0.5 % and held at each strain where another field, temperature and angle dependent dataset was obtained. Again the strain was relaxed and measurements of J c and B c2 taken and were found to agree with the previous results showing the sample remained undamaged.
The very high values of B c2 in REBCO mean it was not possible to measure it directly at low temperatures. The lack of data at high reduced field in the low temperature region also meant it was not possible to determine B c2 using the universal flux pinning scaling curve (as is the case with the Nb 3 Sn sample). To obtain B c2 at low temperatures we first established the universal flux pinning scaling in the high temperature region at θ = 0° (T = 60, 68 and 76 K) using the directly measured values of B c2 . The parameters p and q were then fixed at the values obtained www.nature.com/scientificreports www.nature.com/scientificreports/ from the high temperature data, and the J c data in the low temperature region at θ = 0° (T = 4.2, 20 and 40 K) and all temperatures at θ ≠ 0° were fitted to the universal flux pinning curve allowing B c2 to be a free parameter.
Transport J c data were also taken on a LTS bronze-route Nb 3 Sn wire using the four-probe method with a custom-built probe in an in-house 17 T liquid helium cooled, 40 mm wet-bore, superconducting, vertical solenoid magnet 44 . The field was applied orthogonal to the axis of the wire. Strain was applied to the sample using a Walters spring. Measurements of J c were taken from ε app = −1.16 % to ε app = +0.58 %, at temperatures of T = 4.2, 8, 10, 12, 14 K and various fields chosen such that typically eight in-field measurements were taken at each combination of temperature and strain. Direct transport measurements of B c2 were not obtained for this sample. B c2 was determined from the field at which the pinning force density fell to zero in the universal pinning curve. Figure 3 shows our extensive field B, temperature T and strain ε dependent set of transport J c and B c2 measurements on REBCO where the field was applied orthogonal to the flat surface of the tape and to the Nb 3 Sn wire axis. Figure 4 shows the universal scaling of the normalised pinning force versus the normalised magnetic field for both samples. Additional J c data for the HTS conductor are included in Fig. 4 for different angles θ. The insets show that some of the scatter on the universal curves is associated with κ F p, max 1 2 being double-valued such that its value in tension is not equal to that in compression for the same B c2 , where we have taken m = 2 and www.nature.com/scientificreports www.nature.com/scientificreports/ The bimodal chain model developed for zero field data considers the tape as a chain of domains A and B with relative domain fractions f and (1 − f ) respectively 33 . Under strain, the superconducting properties of one domain increase while those in the other domain decrease. At the highest tensile or compressive strains, J c of the tape is dominated by just one of the domains, namely that with the lowest J c . By considering the tape as a 1D twinned single crystal, the model attributes the inverted parabolic nature of the strain dependence of J c to the competition between the two domains with opposite strain dependencies. This interpretation is in contrast to the standard explanation that attributes the inverted parabolic response of J c (ε) in LTS conductors to the intrinsic averaged behaviour of the underlying material.

Results and Analysis
In the analysis here, we distinguish those features in the bimodal model that are not present in models that attribute J c (ε) to a single component. This identifies the emergent properties of J c (ε). The electric field E generated by a bimodal system is given by   www.nature.com/scientificreports www.nature.com/scientificreports/ with single crystal data 31 . In general, when f < 0.5, as in Fig. 5, ε ε < = J J peak cA cB , taking lower values at lower temperatures, and κ F p, max 1 2 is lower in tension than compression for the same value of B c2 . When f = 0.5 the behaviour of J c is indistinguishable from homogeneous models and ε ε = = J J peak cA cB . When f > 0.5 then ε ε > = J J peak cA cB , taking higher values at lower temperatures, and κ F p, max 1 2 is higher in tension than compression for the same value of B c2 . We conclude that if f ≠ 0.5, ε peak is field, temperature and f dependent which cannot be accounted for by models where measurements are attributed to an averaged or homogeneous underlying material. Also, κ F p, max 1 2 is not a single valued function of T c (ε).
We now calculate approximate values of f and ε = J J cA cB for the HTS and LTS samples by deriving an analytic form for ε peak . For small changes in strain, we can take J ci to have a linear strain response which is equal and opposite in each domain where g is a function of temperature and field calculated by taking a first order Taylor expansion of Eq. (1) in strain about ε JD = 0 %. Equation 4 has the form that follows from the assumption that the field and temperature dependence of J ci in both domains is the same. ε peak is then calculated as the turning point of a second order Taylor expansion of Eq. (3) about ε JD = 0 % to give ) and N 0 is the index of transition at ε JD = 0%. There is typically ~10 % difference between the analytic Eq. (5) and the numerical results in Fig. 5. The functional form of g is dependent on the parameterisation of B c2 which is different for the HTS REBCO and LTS Nb 3 Sn samples. For the REBCO sample, B c2 is parameterised as c2 c c , and s and w are constants. The resulting equation for g is  www.nature.com/scientificreports www.nature.com/scientificreports/ Here we concentrate on identifying and characterising the signature for emergent behaviour, namely the field, temperature and angular dependence of ε = ε peak . We identify the position of the peak by simply fitting the data to a parabola over small strains about the peak. Changes in ε peak caused by thermal expansion are at least an order of magnitude smaller than the variations reported here. The thermal expansion of the REBCO and Nb 3 Sn are determined by the CuBe sample holders because of their large cross sectional areas relative to the samples and are <0.018 % and <0.0005 % respectively 46 . Furthermore, the REBCO sample is constrained by the sample holder in two dimensions so the opposite strain dependencies of the critical parameters in the two directions mean that any effect of thermal expansion on ε peak is further reduced and can be ignored 33 . Figure 6 shows the field and temperature dependence of ε peak and the insets of Fig. 4 show the double-valued behaviour of κ F p, max 1 2 for both samples as expected from bimodal behaviour. In the calculation of g the parameters n, p, q, B c2 (0, 0), T c (0), s and ν are taken from the experimental results, whereas m = 2 and w = 2.2 follow the work of Taylor 9 . Figure 7 shows ε peak against ε where the intercept is ε = J J cA cB and the gradient is used to calculate f. The size of the error bars is predominantly associated with uncertainty in N 0 . The data taken in pool-boiling mode at 4.2 K were omitted from this analysis for both samples, due to large uncertainties in .
The value of f is within the range of those determined from XRD measurements 47 given that values of f and  38 . We note that were the canonical theoretical explanation for the strain dependence of Nb 3 Sn to apply (or if f ~ 0.5), ε peak would be independent of field and temperature and the dashed line in Fig. 7b would be horizontal.
In this paper we have analysed the relatively small inverted parabolic strain range. At large strains, we find convex behaviour in our data that is also in all our numerical calculations. We also find asymmetry in J c (ε), that can be reproduced in our calculations by including a different strain sensitivity of J c (ε) along the a-and www.nature.com/scientificreports www.nature.com/scientificreports/ b-directions 48 . Fitting our data over a larger strain range with multiple components and with different strain sensitivities, introduces more free parameters and will be the subject of future more specialised technical papers.

Discussion
The bimodal chain model was originally shown to be consistent with the properties of REBCO coated conductors in zero field. Here we have developed it to describe the in-field behaviour of REBCO. Strikingly we have discovered that emergent behaviour also occurs in bronze route Nb 3 Sn as shown by the inset of Fig. 4b and in Fig. 6b. Since the strain dependence of the superconducting materials properties, the upper critical field B c2 (ε), the Ginzburg-Landau parameter κ 1 (ε) and the critical temperature T c (ε), are all derived from J c (ε) they must also be considered weakly-emergent.
The model can explain many of the 'anomalous' features of HTS materials in the literature. The field and temperature dependencies of ε peak found in published datasets can be explained by values of f < 0.5 14,49,50 and f > 0.5 51 . The large variations in ε peak between different coated conductors measured in the same experimental setup 16 can also be explained by differences in f caused by the high oxygen mobility at low temperatures in REBCO 52,53 that also is strain-sensitive 48 . Coated conductors have been manufactured using the Inclined Substrate Deposition (ISD) technique that produce a crystallographic orientation of the ab-plane that is rotated by 45° so the [110] direction is along the direction of current flow 30 . In these types of tapes, the strain dependence of both twinned domains is similar so, as with Bi 2 Sr 2 Ca 2 Cu 3 O x conductors which also have unimodal strain behaviour of J c (ε) 54 , there is no competition between the domains 32,55 and it leads to a weak monotonic strain dependence for J c . There is also additional evidence in the literature for bimodal behaviour in other LTS materials as evidenced by the double-valued behaviour of κ F p, max 1 256 .
A deeper understanding of J c (ε) will leverage better strain performance in high field magnet systems through innovative processing of conductors and/or magnet coils. Detwinning HTS materials is already underway to www.nature.com/scientificreports www.nature.com/scientificreports/ improve strain tolerance of conductors 48 . We suggest that aligning tetragonal Nb 3 Sn may similarly also provide increases in J c . While the Nb 3 Sn grains in the bronze-route wire reported here are nearly randomly oriented 57 , in Restacked Rod Processed (RRP) and Powder In Tube (PIT) Nb 3 Sn, partial texturing in the <100> and <110> directions respectively occur 58 . We suggest that fabricating conductors that are strongly textured, particularly if high angle grain boundaries could be removed as in the HTS conductors, would be of great interest to test the model presented here further and possibly to achieve much higher J c at all strains. At present J c in Nb 3 Sn in high fields is less than 1 percent of theoretical limits 59 . The increased technological use of hydrostatic pressure at high temperatures to improve J c in both LTS 60 and HTS superconductors 61 may encourage using additional strain while operating magnets 62 and/or innovative means of applying anisotropic stress during conductor or coil processing heat-treatments to encourage the growth of aligned HTS, or aligned tetragonal Nb 3 Sn. While heat-treating coils, one could simply use mechanical stress directly. However high temperature processing more suitable for industry may include putting physical inserts with different thermal expansion coefficients to the coils in say the bore of the coils and removing them after the heat-treatment, or even using electromagnetic stress, produced by putting current through the copper of the coil conductor.
Such understanding of J c (ε) also helps identify the intra-and intergranular microscopic origins of the component parts with opposite strain dependencies in HTS and LTS materials. In HTS, extensive single crystal data directly identifies intragranular properties as one source of competing strain dependencies in the twinned tapes. Although stoichiometric A15 materials can be cubic, technological high field superconductors are generally off-stoichiometric and anisotropic. Anisotropic strain dependencies in Nb 3 Sn are demonstrated by (the limited) single crystal data that show along the (001) direction dT ci /dε (100) = 1.63 K% −138 and, similarly to HTS, the hydrostatic strain dependence is much smaller, dT ci /dε (hydro) = 40 mK% −1 34,63 . Given that both REBCO and Nb 3 Sn tapes show that minimising deviatoric strain increases J c 64 , we conclude that competing intragranular components are important in both REBCO and Nb 3 Sn. All polycrystalline A15 (including Nb 3 Sn) superconductors measured to date 65 (as well as REBCO reported here, and the superconducting ductile alloy NbTi 66 ) have J c (ε) that reaches its peak value when the intrinsic strain is close to zero. As the number of different A15 superconducting materials showing this peak continues to increase, it becomes increasingly untenable that this is because of a fortuitous coincidence between the Fermi energy and a peak in the density of states 35 . Nevertheless for decades, researchers have assumed that measurements on such polycrystalline materials have provided the angularly averaged properties of these materials 34 without adequate single crystal data. Although the primary origin of the emergent behaviour in polycrystalline Nb 3 Sn is probably associated with the grains and grain boundaries (discussed below) 59 , the canonical explanation for the fundamental inverted parabolic strain dependence of T c itself can be challenged since the calculations have only been completed for stoichiometric A15 compounds rather than for the computationally more demanding off-stoichometric, alloyed materials 67 found in technological wires, and although there is good long-standing evidence for A15 superconductors being strongly coupled BCS superconductors 68 , the Uemura plot presents the possibility that A15 materials may be non-BCS superconductors 69 . For non-BCS superconductors, such as the HTS materials, one simply cannot properly address T c (ε) because there is no reliable explanation for the fundamental mechanism causing the superconductivity.
In polycrystalline Nb 3 Sn, at J c , dissipation occurs because of flux flow along the grain boundaries where the local superconducting properties are degraded 14  Hence the effect of Poisson's ratio will give rise to intergranular contributions to J c in polycrystalline materials with opposite strain dependencies because under either compressive or tensile strain, the width of some grain boundaries will increase whilst others will decrease, which will change the coupling between neighbouring grains. Since intergranular superconducting properties are determined by both the grain boundary itself and the grains on either side of the boundary, in general, strain dependencies of both intra-and intergranular components will be important in Nb 3 Sn. Such general considerations of the channels along which flux flows at J c (ε) (e.g. grain boundaries), provide a explanation for why optimum properties are so commonly observed in polycrystalline A15 superconductors close to the unstrained state. Furthermore, to understand the measured properties correctly and to characterise them accurately, these properties must be considered emergent -they are not the angular average of the underlying material, nor are they the properties associated for example with one particular (e.g. the most) degraded region of the material. In HTS materials, it is not clear yet whether low angle grain boundaries or twin boundaries are the location where the flux first moves at J c , whether flux moves after depinning within channels over-populated by pins within the grains 14 , or after depinning from single pinning sites within grains 5 . Hence whether intra-and intergranular properties must both be considered in HTS, as is the case for Nb 3 Sn, is still open.
To date, the standard literature has continued to describe J c in closed form using Eq. (1) even after adding the strain dependence 7,9,10 . However we have found that the mathematical approach required to extend the range of properties included in the functional form of J c has depended on whether the new properties are primary or emergent. To achieve an accurate description of strain dependencies, that includes the field and temperature dependence of ε peak , a different mathematical approach has been required. For as long as only primary properties (B and T) were included, J c was a scaling law of closed form given by Eq. (1). Adding the strain dependence meant replacing the scaling law expression for J c by a transcendental equation (Eq. (3)) and restricting the scaling law to be a description of the field and temperature dependence of the component parts alone. Hence the argument that the monotonic strain behaviour (Eq. (4)) of the underlying components is qualitatively different to the inverted parabolic behaviour of the overall J c (Fig. 5) has been supported by the change in the structure of the mathematics describing J c . This underpins Anderson's rewording of the clichéd description of emergence: 'the total is… different to the sum of the parts' 71 . There is also a change in the important relevant length scale between the primary and emergent properties. The size of the basic building block that determines the field and temperature dependencies of J c (B, T) typically has dimensions of a few times a characteristic superconducting length-scale (e.g. the coherence length, the penetration depth or the flux-line-lattice) depending on the nature of the pinning.
The properties of a single grain boundary of an LTS material or a single domain for an HTS material are sufficient to characterise the functional form J c (B, T) for the whole material. Whereas the basic building block needed to describe the strain dependencies of J c (ε) for the whole material is determined by the microstructure. We need a few competing domains with opposite strain dependencies to understand REBCO or a few competing grains and grain boundaries to understand Nb 3 Sn. Describing the weakly-emergent strain-dependent properties of high field superconductors does not require a very detailed understanding of the complexity of flux pinning, or very precise exponents for the scaling law, any more than describing emergent behaviour in biological systems needs a very detailed understanding of the complexity of the individual insects or birds. In this work, the conclusions and insights into the effect of strain are not sensitive to the precise values of the exponents used in the scaling law (Eq. (1)). In both the superconducting and biological systems, an additional set of equations (e.g. (Eqs (3 and 4)) or local rules leads to a description of the emergent property or overall behaviour.
Amongst the materials physics community, superconductivity is often considered to be the example par excellence for emergence. At the critical temperature (in zero field), the sea of normal electrons collectively condense into Cooper pairs 72 and bring with them the property of zero resistance 73,74 . The high magnetic field properties are best described by Ginzburg-Landau equations 8 which include a macroscopic wavefunction as a ground state and together with Abrikosov's insights 75 , eventually led to the concepts of flux quantisation and flux pinning. We have taken Eq. (1) that describes the field and temperature dependence of the whole material, as the starting point in this paper. However, from a starting point that begins with the sea of normal electrons, even when Eq. (1) is only applied to the flux pinning in the components of these high-field superconductors, it describes emergent behaviour. This has similarities with the classification of the living things considered before. The sociologist considers the behaviour of the individual birds and ants primary, and the behaviour of the flocks and swarms emergent. However, the chemist considers the behaviour of molecules primary, and that of the individual birds and ants emergent 71 . We suggest our work describes the properties of an interesting technologically useful solid-state material that can provide a useful case-study for weak-emergence. The properties of the components are well-defined and relatively simple mathematically (Eq. (4)), as is the relationship giving the competition between the component parts that leads to the overall behaviour (Eq. (3)).

concluding comments
While the approximations that consider high field superconductors as simple homogeneous materials can provide useful engineering parameterisations of J c for magnet design, particularly for LTS materials where the field and temperature dependence of ε peak is relatively small, we have shown here that this does not describe the underlying science. We have made the observation that ε peak varies with field and temperature in both an HTS and a LTS conductor and conclude that any description of similar high field superconductors that attributes the reduction in J c under either compressive or tensile strain to the intrinsic averaged underlying strain dependence of any simple component of these materials will not explain the changes in ε peak reported here. The evidence for the emergent behaviour in the HTS tape presented in this work follows from a detailed analysis of J c (ε) data and a comparison with single crystal data. Although the strain-dependence in single crystals of Nb 3 Sn is far less detailed than that reported for HTS materials, the A15 single crystal experimental data available do not provide support for intrinsic parabolic behaviour in Nb 3 Sn, with a peak in J c observed near zero-intrinsic strain. Hence in addition to the lack of experimental evidence from single crystals supporting the canonical explanation for T c (ε), we add the experimental data in Fig. 6 and the analysis presented here, to make the prima facie case that emergent behaviour also occurs in Nb 3 Sn wires.
It will be a huge challenge to measure and understand the underlying competing components in high field superconductors. Analysis will need to include percolative current flow, and measurements will be required of the anisotropic strain dependence of the superconducting properties of single crystals. In HTS materials, the artificial pinning centres that have produced the highest J c values will further complicate understanding the anisotropy of the materials 76 . In LTS materials, there is an obvious need for detailed experimental studies of the anisotropy of off-stoichiometric and alloyed tetragonal single-crystals. To understand LTS materials will also require detailed local measurements of grain boundaries on the scale of the coherence length which will be very difficult. We probably need to develop new tools and new types of experiments for investigating the grain boundaries of polycrystalline metals and may for example use some aspects of the approach that used electron-beam-induced current to look at the electronic properties of grain boundaries in semiconductors, to achieve this 77 . For as long as the flux pinning law (Eq. (1)) included only magnetic field and temperature dependencies, it could be considered a primary law that described averaged property dependencies and fitting parameters. Adding the requirement for the strain dependency of J c meant that the scaling law had to be restricted to describing component parts, and a new transcendental equation for the overall behaviour of J c was required. This restructuring reminds us of the concern that labelling the properties of an object as emergent, and hence qualitatively different to those of its components, is a subjective judgement. This concern becomes particularly problematic as we consider biological systems and properties such as life or consciousness where agreeing on the essential properties of the components and the overall system is not straightforward 78 . Here we have found that the inclusion of an emergent property is flagged by both a change in the structure of the mathematics and in the important length scales.
We suggest that describing emergence mathematically is not solely a triumph of aesthetics. Using the best category of law (primary or emergent) for the relevant degree of complexity can improve both utility and understanding. The new high field mathematical framework described provides the technological utility of a more accurate description J c . It also provides a better understanding of how the strain dependence of J c (ε) arises. We suggest that this understanding of J c (ε) as emergent will aid magnet engineers trying to improve high field superconducting materials under strain. This work may also provide a well-defined and simple case-study that can help the broader scientific community develop the language and taxonomy of emergence.