Implications of the strain irreversibility cliff on the fabrication of particle-accelerator magnets made of restacked-rod-process Nb3Sn wires

The strain irreversibility cliff (SIC), marking the abrupt change of the intrinsic irreversible strain limit εirr,0 as a function of heat-treatment (HT) temperature θ in Nb3Sn superconducting wires made by the restacked-rod process (RRP®), is confirmed in various wire designs. It adds to the complexity of reconciling conflicting requirements on conductors for fabricating magnets. Those intended for the high-luminosity upgrade of the Large Hardon Collider (LHC) at the European Organization for Nuclear Research (CERN) facility require maintaining the residual resistivity ratio RRR of conductors above 150 to ensure stability of magnets against quenching. This benchmark may compromise the conductors’ mechanical integrity if their εirr,0 is within or at the bottom of SIC. In this coupled investigation of strain and RRR properties to fully assess the implications of SIC, we introduce an electro-mechanical stability criterion that takes into account both aspects. For standard-Sn billets, this requires a strikingly narrow HT temperature window that is impractical. On the other hand, reduced-Sn billets offer a significantly wider choice of θ, not only for ensuring that εirr,0 is located at the SIC plateau while RRR ≥ 150, but also for containing the strain-induced irreversible degradation of the conductor’s critical-current beyond εirr,0. This study suggests that HT of LHC magnets, made of reduced-Sn wires having a Nb/Sn ratio of 3.6 and 108/127 restacking architecture, be operated at θ in the range of 680 to 695 °C (when the dwell time is 48 hours).

the fractions of Sn and Nb and decreasing that of Cu in the wire design [6][7][8] . It results in Nb 3 Sn occupying about a third of the wire's cross-section 9 , with a near stoichiometric composition [6][7][8][9] . Both of these features are needed to obtain very high J c . On the other hand, during HT, individual filaments coalesce into one solid Nb 3 Sn tube inside each of the wire sub-elements, of a diameter in the range of 35 to 100 μm depending on the wire architecture 7,8 .
Whereas J c values in excess of 3,000 A/mm 2 at 12 T and 4.2 K obtained in RRP Nb 3 Sn wires are sufficient for HL-LHC, magneto-electrical instabilities that stem from magnetic-flux and electric-current fast redistributions in the conductor during current ramping could be challenging [10][11][12][13][14][15][16][17][18] . These instabilities, referred to as magnetization and self-field instabilities, respectively, generate local heat and temperature rise that can cause magnets to quench at currents significantly below the conductor's critical surface of J c versus magnetic field B at low and moderate field ranges [10][11][12][13][14][15][16][17][18] . Reducing the size of Nb 3 Sn sub-elements can mitigate these instabilities significantly, and a sub-element size around 20 μm or less is desirable 1 . However, size reductions below 50 μm in RRP wires have resulted in a dramatic drop of J c 8, 19 . In addition, such wires still need further development to become technological conductors that can be readily fabricated in piece-lengths sufficient for applications. Furthermore, such sub-element reduction may impose additional limitations on HT temperature and duration to avoid breaching distributed Nb barriers that would otherwise result in a significant decrease of the wire's residual resistivity ratio RRR 1 , a parameter that is also important to remedy conductor instabilities [11][12][13][14]16,17 .
Magnetization and self-field instabilities may be contained if RRR of the conductor's Cu component is sufficiently high 14,16,17 . Basically, magnet quenching can be prevented if disturbance-generated heat is evacuated to the surrounding helium bath fast enough through Cu. The thermal conductivity of Cu improves by a factor of about 3.4, in the temperature range 1-10 K, when its RRR is increased from 30 to 100 20 . Such an improvement may be useful for a more effective evacuation of heat away from the magnet 21 . Extensive empirical and simulation studies of wire instabilities suggested that RRR be at least 100 in the final conductor form (once cabled and reacted as part of the magnet) to mitigate instability effects 14,16,17 . This translates to an RRR in undeformed conductor (i.e. before cabling it) of at least 150.
This requirement RRR ≥ 150 was achieved, first, by applying less aggressive HTs that promote RRR but at the expense of J c 14,16,17,22 . Thereafter, a 6% reduction of the amount of Sn in the wire turned out to be very effective for meeting the RRR requirement 8 . Indeed, a change of the ratio Nb/Sn from 3.4 in early RRP billets (referred to as standard-Sn billets) to 3.6 in new ones (reduced-Sn billets) was a beneficial adjustment in the wire design 8 . Even though J c of reduced-Sn billets decreases due to the reduction of the amount of Sn in the wire, it can be recovered partially by increasing the HT temperature typically from 640-650 °C for the standard-Sn billets to 665-680 °C for the reduced-Sn billets without compromising the RRR benchmark.
The sub-element size for the wires to be fabricated for HL-LHC has been set to a compromise value of about 55 μm to maintain a sufficiently high and homogeneous J c 2,23 , further underscoring the need to maintain high RRR to mitigate wire instabilities.
In a recent report, the U.S. high-luminosity LHC accelerator upgrade project (US HL-LHC AUP) added a requirement "ε irr,0 > 0.25%" to the strand design criteria for the fabrication of quadrupole magnets 23 , ε irr,0 being the intrinsic irreversible axial-strain limit where irreversible effects first appear in the behavior of the conductor's transport critical current I c as a function of axial strain ε to which it is subjected. The benchmark value of 0.25% is actually empirical, based on earlier strain characterizations of Ti-doped RRP Nb 3 Sn wires 41 . In effect, even higher ε irr,0 values are better to help reduce the complexity of the mechanical structures and tooling needed for stress and strain management in magnets.
We recently showed that ε irr,0 depends strongly on HT conditions in RRP Nb 3 Sn standard-Sn billets either doped with Ti or Ta. We found that ε irr,0 undergoes a precipitous and large change as a function of HT temperature θ, called the strain irreversibility cliff (SIC) 46 . As we discussed in 46 and will show in great detail herein, SIC too imposes certain conditions on HT with competing requirements vis-à-vis those favorable to RRR. As such, SIC adds to the complexity of the trade-offs among J c , RRR, and sub-element size benchmarks. Therefore, studies of the conductors' electromechanical properties should be combined with investigations of J c and RRR as a function of conductor design and HT to find optimal parameters for magnet conditioning. In order to evaluate the practical implications of SIC comprehensively, such approach is essential.
In this paper, we build on our previous strain characterization of standard-Sn billets RRP 13711-2 (Ta-doped) and 11976-1 (Ti-doped) by adding two more billets, RRP 14943-2a and 14984, both Ti-doped (Table 1). Billet 14984 has a reduced-Sn content. We will put more emphasis on it as it is a pre-production billet for HL-LHC. We investigate if such a conductor exhibits the SIC behavior and how it is affected by the Sn content. Billet 14943-2a is a standard-Sn, used for comparisons and for gaging reproducibility of the results with respect to billet 11976-1 that has the same design but a slightly smaller diameter (Table 1). Furthermore, we expand the study to encompass RRR measurements on all four billets. We investigate how the HT temperature θ affects ε irr,0 (and SIC), I c , and RRR of a given conductor, and define the range of θ that is suitable to fulfil the competing requirements on the multiple parameters for each of the conductors and the role played by the content of Sn and the doping element.
Each of the four billets has 108 Nb 3 Sn sub-elements distributed around 19 Cu sub-elements at the billet center (design 108/127). Additional details are provided in Table 1. The HT (made in vacuum) consists of two pre-stages at 210 °C for 72 hours and 400 °C for 48 hours to mix Cu and Sn, followed by a final stage at temperature θ for

Results
We used a Cu-Be Walters spring to strain samples in-situ [57][58][59] . Information on the apparatus is provided in the Methods section. For each sample (of helical form), we determined I c values for three sample locations, each being one full sample turn (or segment) about 8 cm long. Measurements were made while the sample was immersed in liquid helium at a temperature of 4.04 to 4.07 K and subjected to an external magnetic field of 15 T provided by a superconducting solenoid. For each billet, we measured one to three samples (i.e., three to nine segments) per HT. For this study, I c (ε) measurements were made on 94 samples in total. Values of I c were determined at the electric field criterion E c of 0.1 μV/cm.
Details of the RRR apparatus are also given in the Methods section. Measurements of sample resistance R as a function of sample temperature T were made to determine RRR, defined as the ratio of R at 293 K to that at 18 K. For each billet, we measured three samples per HT. For this study, RRR measurements were made on 81 samples in total. Values of RRR in the literature are generally quite scattered, so the results we present here may not be representative of those of a large number of billets of a given design.
Expanded uncertainties (k = 2) due to random effects in estimating θ, I c , ε irr,0 , and RRR were 3 °C, 2%, 0.03% strain, and 30%, respectively. The uncertainties for I c , ε irr,0 , and RRR are based on type A evaluations of uncertainty, whereas the uncertainty for θ is based on type B evaluation of uncertainty 60 .
Effects of heat-treatment temperature on ε irr,0 and RRR. Examples of I c (ε) data are provided in  46 for billets 11976-1 and 13711-2. Strain applied to the sample is increased incrementally until I c reaches its peak value I c-max at strain ε max that compensates for the Nb 3 Sn compressive pre-strain. The latter develops during cool-down of the sample from θ to 4 K, and originates from the mismatch of thermal expansion coefficient of Nb 3 Sn with respect to those of the other wire components and that of Cu-Be spring material to which the sample is soldered. Around ε max , we unload strain partially and remeasure I c to check its reversibility with strain. This operation is repeated multiple times while increasing strain gradually until I c is driven close to zero. From the "loaded" and "unloaded" (solid-and empty-symbol) curves thus obtained, we determine the irreversible strain limit ε irr that produces the first splitting of the two curves (see details in the Methods section) 42 . The partial-unloading step is kept constant at 0.09% throughout the experiments; it is small enough to minimize I c increases upon unloading (for ε < ε irr ) due to the three-dimensional strain effects 43,61 , and large enough to reveal I c (ε) irreversibility. Locations of ε max and ε irr are marked by arrows in Fig. 1(a-c).
Our values of ε max and ε irr are artificially high because the thick Cu-Be spring dominates the thermal contraction of the assembly (spring + sample) and puts an additional compressive pre-strain on Nb 3 Sn upon cooling to 4 K. These values can also change slightly from sample to sample depending on how tight the sample is made on the spring before soldering. Nevertheless, I c versus intrinsic strain ε 0 (=ε − ε max ) is not affected by the spring material's differential thermal contraction or by the sample mounting 35,62 . Therefore, the same should hold for the values of the intrinsic irreversible strain limit ε irr,0 (=ε irr − ε max ). Absolute values of ε max (and ε irr ) provided herein should not be taken as a direct measure of strain margins for the wires. In fact, the actual values of ε max for RRP wires are rather small due to the high Nb 3 Sn fraction as compared to moderate-J c Nb 3 Sn conductors 63 . Thus, ε irr,0 values are appropriate to gage the strain resilience of the wires as investigated here.
Values of ε irr,0 are displayed in Fig. 1(a-c) in red for their corresponding billet and HTs. The complete dependence of ε irr,0 on θ is depicted in Fig. 2(a) for the standard-Sn, Ti-or Ta-doped wires, and in Fig. 2(b) for the standard-or reduced-Sn, Ti-doped wires. Each point represents an average value of ε irr,0 over one to three samples (i.e., three to nine segments) per HT. For all four wires, ε irr,0 exhibits a precipitous and large change with θ, indicating a transition of Nb 3 Sn from a highly brittle state, where irreversible effects start as soon as Nb 3 Sn is subjected to a tensile strain of any measurable amount, to a more strain-resilient state where changes of I c (ε) remain reversible up to a tensile strain close to 0.4%. This behavior is the strain irreversibility cliff (SIC) 46 ; it is confirmed here for the four RRP Nb 3 Sn billets. These results clearly show that θ (or HT schedule in general) has a major influence on ε irr,0 .
For the four billets, values of ε irr,0 on the SIC plateau are above the required value of 0.25%. The cliff extends over a narrow temperature range of ≈23 to 29 °C, and its bulk part (excluding the tail) is much steeper as it occurs  The intrinsic irreversible strain limit ε irr,0 (=ε irr − ε max ) has a strong dependence on the heat-treatment temperature θ. The sample was loaded and partially unloaded (by constant axial-strain steps of about 0.09%) to obtain the "loaded" and "unloaded" I c (ε) curves, represented by solid and empty symbols, respectively. Corresponding loaded and unloaded points are labelled by unprimed and primed letters, respectively. ε irr is defined as the applied strain that produces the first splitting of these two curves. ε max is the applied strain that compensates for the sample's pre-compressive strain. www.nature.com/scientificreports/ 14 °C for the reduced-Sn wire ( Fig. 2(b)) [see horizontal black arrows in Fig. 2(a,b)]. Also, the height of the cliff is lower for the standard-Sn, Ta-doped, or reduced-Sn, Ti-doped wires [see vertical black arrows in Fig. 2(a,b)]. Nevertheless, for the reduced-Sn wire, ε irr,0 improved further and precipitously when θ is increased beyond 705 °C [see orange arrow in Fig. 2(b)], revealing a double-cliff structure for this particular billet, though the second step is much smaller than the main cliff height. We do not know yet if this double-cliff structure is typical of reduced-Sn billets. These differences from billet to billet may provide clues regarding the origins of SIC. This paper is principally focused on treating the implications of SIC on applications.
The results of RRR(θ) are depicted in Fig. 3, where each data point represents an average value of RRR over three samples. Figure 3 illustrates the strong decline of RRR as θ is increased, especially for the standard-Sn billets. The red line delimits RRR = 150, and is crossed at θ ≈ 639, 656, 664, and (approximately projected) 745 °C for billets 14943-2a, 11976-1, 13711-2, and 14984, respectively. (This temperature is noted below as θ RRR ). The reduced-Sn billet shows a significantly higher RRR that stays above 150 for an extended range of θ as compared to the standard-Sn billets.
Electro-mechanical stability (EMS) criterion. As discussed in 46 , the narrowness of the θ range where SIC occurs may have challenging implications for the heat-treatment of large magnets. Reducing θ to promote the strand's magneto-electrical stability will be at the expense of the strand's strain properties if ε irr,0 is not positioned away from the SIC tip somewhere on the SIC main plateau. The magneto-electrical (or electrical for simplicity) and mechanical requirements are in conflict, and we need to find HT conditions that achieve them both. In our analysis, we build on the requirements RRR ≥ 150 and ε irr,0 ≥ 0.25%, but, for the latter, we put more emphasis on staying away from the cliff to ensure that the inevitable temperature gradient across the magnet during heat-treatment and furnace-temperature imprecisions do not produce weak ε irr,0 anywhere in the magnet. This analysis is flexible enough to accommodate other applications if RRR requirements are different. www.nature.com/scientificreports/ To link RRR(θ) and ε irr,0 (θ) results for analysis, we define θ RRR as the temperature where RRR crosses the required minimum value (150 in the case of LHC magnets) and θ Cliff as the temperature where the precipitous drop of ε irr,0 starts to occur (tip or onset of SIC). The choice of θ must be such that the wire fulfills both requirements for electrical stability (RRR ≥ 150 for LHC magnets; it could be a different value for another application) and mechanical stability (ε irr,0 at the top of SIC). First, billet properties must be such that θ Cliff is lower than θ RRR with enough margin; then, the operator must choose θ between these two values. We capture these recommendations in what we call the electro-mechanical stability (EMS) criterion as:

RRR Cliff
where δθ is introduced to account for uncertainties in θ arising from possible inhomogeneity and inaccuracy of furnace temperature. If a wire (considered for fabricating a magnet) meets this requirement, then θ must be chosen such that RRR Cliff thus setting the allowable window for best HT. We define the EMS criterion by both equations (1) and (2), even though fulfilling Eq. (2) implies the same for Eq. (1). We wanted to indicate that some billets may be intrinsically limited to fulfil Eq. (1), in which case fabricating a magnet from such billet should not be even considered (let alone heat-treating it). We believe the EMS criterion is a valuable tool to inspect billets before considering their use in magnets. We illustrate this concept in Figs 4-6 for the four wires, for the LHC magnets.
In Fig. 4(a), ε irr,0 (θ) and RRR(θ) are plotted together for the standard-Sn, Ti-doped billet 11976-1 (the arrows near the bottom point to the Y-axis corresponding to each curve). The areas shaded in gray delimit the domains where θ < θ Cliff (unsatisfactory mechanical properties) and θ > θ RRR (unsatisfactory electrical properties). The hatched areas in the remaining window (in Fig. 4a) represent temperature safety margins of a width δθ ≈ 5 °C on each side to ensure that conductor performance is sufficiently away from the SIC tip and away from getting below RRR of 150. Thus, the residual domain between the hatched areas (pointed to by an arrow in Fig. 4(a)) defines the allowable temperature window for an optimal HT, as defined in Eq. (2). In this case of billet 11976-1, the allowable θ is 648 ± 3 °C; a very narrow range that might be difficult to apply for heat treating large magnets (4 to 8 meters long in the case of HL-LHC). The value of 5 °C for δθ is arbitrary and used here for illustration purposes. Actual value of δθ can be adjusted depending on the quality of the furnace used and on the size of the magnet. It can be also adjusted to reflect θ Cliff and θ RRR variations among a large number of billets of a given design (when such statistical data are available).
In Fig. 4(b), the dependence of I c-max on θ is shown for same billet 11976-1 (see ref. 46 for more details on this dependence and on the choice of I c at this particular strain reference ε max ). The gray and hatched areas of Fig. 4(a) are pasted in Fig. 4(b) to evaluate the consequences on I c of the EMS criterion's HT restrictions. It appears that staying within the allowable HT window means that this wire will be used at approximately 10% below its full I c potential. Figure 4(b) reveals the trade-off between the electro-mechanical and transport properties for this www.nature.com/scientificreports/ particular billet, for a HT conducted at the optimal θ (=648 °C) for a duration of 48 hours. We will comment on the effect of HT duration later.
Analysis for the standard-Sn, Ta-doped billet 13711-2 is depicted in the same way in Fig. 5. For this billet, the allowable HT window is 658 ± 1 °C (Fig. 5(a)). It is located at a temperature 10 °C higher than for billet 11976-1 but is narrower, making it even more difficult to apply such HT. As shown in Fig. 5(b), this wire will be at approximately 9% below its full I c potential. The case of the standard-Sn, Ti-doped billet 14943-2a illustrated in Fig. 6(a) is the worst. The HT window is too small, even smaller than the safety margin δθ. Hence, the EMS criterion is not met and, as such, there is no temperature window to secure an optimal HT (at 48-hour dwell time) that will satisfy both the electrical and mechanical requirements. Therefore, we suggest that such a billet be discarded to avoid fabricating magnets with poor properties and save valuable funds. This particular billet may not be representative of the standard-Sn billets produced, considering its low values of RRR (see Fig. 3), but is illustrative of the implications of SIC that could possibly happen. Furthermore, existence of outlier billets in a large production volume cannot be excluded.
The narrowness of the allowable HT window for the three standard-Sn billets, dictated by the EMS criterion, is not desirable. It might be a general trend for standard-Sn billets, but we cannot be certain as there is some RRR variability from billet to billet. Nevertheless, alternative HTs with lower temperatures for longer dwell time might render such billets more suitable. By lowering θ below θ Cliff , ε irr,0 will decrease but perhaps will be restored if the dwell time is increased sufficiently. However, we should also consider the dependence of RRR on dwell time. Lowering θ will increase RRR, but increasing the dwell time will tend to decrease RRR. So, it is a matter of finding the right balance between θ and dwell time to achieve high ε irr,0 and high RRR. This deserves a study of its own to supplement the results presented here. Effects on the allowable HT window of a simplified HT scheme that has two stages (instead of three) to control the Sn-Nb-Cu Nausite phase are also worthy of a systematic study 19 . www.nature.com/scientificreports/ For the reduced-Sn billet, in contrast, the situation is far better. As depicted in Fig. 6(b), the allowable θ is 684 ± 56 °C-a very wide range. Moreover, it is centered very close to the optimum value of θ (≈ 680 °C) where I c is maximal (see Fig. 6(c)). Therefore, no trade-off is needed between I c and RRR for such billet except for the lowering of I c due to the reduced content of Sn (see the discussion below). Such a billet seems very suitable. It is fortunate that HL-LHC will use the reduced-Sn RRP billets design.
Comparison of I c-max between the standard-and reduced-Sn billets 14943-2a and 14984 is displayed as a function of θ in Fig. 7. These two billets are well suited for a direct comparison of I c because they have the same non-Cu area ( Table 1). Values of I c-max are the same for both billets for θ ≤ 630 °C. Differences appear above 630 °C and culminate at about 7% around 680 °C in favor of the standard-Sn billet. However, considering that the reduced-Sn billet can be used with no I c trade-off and that the standard-Sn billet may be used but at less than its full I c potential due to the EMS-criterion restrictions, the reduced-Sn billet is in fact the better option even in terms of I c . If higher I c is needed, we could envision a billet design having an amount of Sn reduced by less than 6% compared to the standard-Sn design (for example 4% or so). Such design may increase I c but may also shrink the allowable HT window in comparison to that of billet 14984 (Fig. 6(b)), though we speculate that it may still be wide enough. Also, it might be possible that such a modified billet will have a higher ε irr,0 than billet 14984, somewhere between the values for billets 14984 and 14943-2a ( Fig. 2(b)).
The US HL-LHC AUP in collaboration with CERN decided that HT of magnets be made at 665 °C (for a dwell time <50 hours) to stay away from SIC 23 . Whereas this choice is consistent with what we presented here so far, below we show that an increase of this temperature by about 15 to 30 °C may have some additional benefits.
Searching for useful information beyond the irreversible strain limit. The reversible effects of strain have been studied extensively since the 1970s, once the high sensitivity of Nb 3 Sn properties to axial strain became evident 64 . Comparatively, the strain boundary of these reversible effects, ε irr,0 (or ε irr ), has received very little attention. As for the irreversible effects themselves, they have probably been considered not worth studying beyond using the data to determine ε irr,0 (or ε irr ). This limit is generally considered as the intrinsic strain where www.nature.com/scientificreports/ cracks start to form in Nb 3 Sn. Hence, disregarding data beyond it is understandable, considering that applications should not be designed to exceed this limit 41 .
In a recent report, we showed some evidence that I c degradation rate in the irreversible regime also depends on HT conditions 46 . This is the reason why we conducted measurements of I c (ε) far beyond ε irr for the samples presented here, until I c was driven close to zero as in Fig. 1. In Fig. 1(b), for example, the shape of I c (ε) (loaded curve) for 666 °C has an "elbow" marking a severe degradation of I c that does not start immediately at ε irr . This feature in I c (ε) dependence is also present for several HTs below 666 °C, and more pronounced, but disappears for HTs above 666 °C as in Fig. 1(c) for 746 °C. www.nature.com/scientificreports/ In Fig. 8, we show a comparison of I c (ε 0 ) for HTs done at 656, 666, 681, and 695 °C, for the specific measurements protocol (described above) that we applied in all these experiments. Note that the curves represent raw I c data (not normalized), plotted as a function of the intrinsic strain (ε 0 = ε − ε max ) to eliminate slight differences of ε max from sample to sample. The values and behavior of I c (ε 0 ) are very similar for the three samples reacted at 666-695 °C, even past ε irr,0 , until the elbow feature appears for the sample reacted at 666 °C that distinguishes it from the other two samples. This feature is even more pronounced for the sample reacted at 656 °C, indicating its progressive evolution with lowering temperature θ. The four samples have about the same value of ε irr,0 ( Fig. 2(b)).  www.nature.com/scientificreports/ Disappearance of the severe drop of I c (ε 0 ) for samples reacted at 681 °C and above may possibly indicate an increase of the fracture toughness of Nb 3 Sn that perhaps becomes more resistant to additional cracking and crack propagation (at a given strain) in these samples. Fracture toughness measurements on individual filaments are needed to verify this assumption. These observations suggest that it may be judicious to increase θ for the LHC magnets from 665 °C to somewhere between 680 and 695 °C (when the dwell time at θ is 48 hours) to contain I c irreversible degradation, especially that this increase in θ keeps it still within the allowable HT window (Fig. 6(b)). Even though the design of these magnets should be such that ε irr,0 is not exceeded, this adjustment of θ would provide an extra precaution.

Conclusion
We coupled studies of I c (ε) and those of RRR for various RRP Nb 3 Sn billets heat-treated at different temperatures from 599 to 752 °C for 48 hours. The billets were either standard-or reduced-Sn, doped with either Ti or Ta, of the type intended for use in particle-accelerator magnets for HL-LHC. They all exhibited an abrupt change of ε irr,0 as a function of θ, known as the strain irreversibility cliff. The approach of combining the two studies allowed us to comprehensively assess the implications of SIC on restricting the HT conditions. We introduced the electro-mechanical stability criterion that takes into account both requirements for electrical (RRR ≥ 150) and mechanical (ε irr,0 away from SIC) stability for best outcome. For the standard-Sn billets, fulfilling these conflicting requirements yields a significant narrowing of the allowable HT temperature window that is impractical. On the other hand, the reduced-Sn billets offer a significantly wider choices for HT, not only for ensuring that RRR ≥ 150 and ε irr,0 is located at the top of SIC, but also for containing the strain-induced irreversible degradation of I c beyond ε irr,0 . This study suggests that HT of magnets for HL-LHC, made of reduced-Sn RRP billets having a Nb/Sn ratio of 3.6 and 108/127 restacking architecture, be conducted at a temperature of 680 to 695 °C, about 15 to 30 °C higher than the 665 °C (for a dwell time <50 hours) that is targeted by HL-LHC project.
Finally, we highly recommend investigating whether the effects of transverse compression are also dependent on HT conditions. This will allow the magnet designer to account for effects of HT on conductor resilience against the various stress components in a magnet. Of course, the reversible effects should be taken into account as well, and be measured or projected accurately through I c parametrization within the reversible strain regime (ε 0 ≤ ε irr,0 ) as a function of magnet operating conditions of strain, temperature, and magnetic field 37 . A spreadsheet was created to facilitate these calculations 65 .

Methods
Apparatus for I c (ε) measurements. Transport measurements of I c as a function of applied axial strain ε were conducted by use of a Walters spring made of Cu-Be (see Fig. 9(a)) 57-59 . This spring device had four turns with a T-section profile, designed such that the spring behaves elastically over a wide strain range from −1% to +1% 58,59 . By twisting one spring end with respect to the other end, the spring's outer surface where the sample was attached can be subjected to either a tensile or a compressive axial strain [57][58][59] . The sample was first mounted on a stainless-steel mandrel for heat treatment, then transferred onto the spring carefully and soldered to it along its full length at ≈200 °C. The sample length prepared (~2.5 m) allowed for multiple spare turns that we cut and used as current splices (visible in Fig. 9(a)). They were soldered to the sample ends to facilitate current transfer into the sample with minimal contact resistance. Multiple pairs of instrumentation leads were attached to the sample to monitor voltage V during ramping of electric current I through the sample. Each of the three main pairs covered www.nature.com/scientificreports/ one of the sample turns (≈8 cm long each) that were on the spring turns. Values of I c were determined from V-I characteristics at various electric-field criteria E c . Data shown here are for 0.1 μV/cm that is a more commonly used criterion.
To extract the values of ε max and ε irr , a polynomial function (at varying orders) was fitted to the loaded I c (ε) data excluding significantly degraded data points (i.e. strain points that yielded a degradation in excess of 5% or so upon strain partial-unloading). The roots of this polynomial function were then found through derivation, and the realistic root value was identified as that of ε max . The high density of acquired data points around ε max (see Fig. 1) made identification of the realistic root very simple. For ε irr , the residuals (I c-measured /I c-predicted − 1) for the unloaded points-I c-predicted being I c calculated by use of the polynomial fitting function assuming no I c irreversible degradation-were determined as a function of ε. Points with residuals around 1% were then extrapolated to find the strain value with a residual of 0. This strain, at the onset of I c irreversible degradation, was identified as ε irr . It corresponds to strain that caused the first splitting of the loaded and unloaded curves (see Fig. 1). More details are provided in 42 . Apparatus for RRR measurements. Samples for resistivity measurements were mounted on the apparatus shown in Fig. 9(b). This RRR apparatus was designed to accommodate six straight samples mounted in series. Pressure contacts were used to connect the current and voltage contacts to each sample. Voltage-tap pairs on each sample were separated by 10 cm. The probe was inserted into a cold cryostat and lowered very slowly to gradually cool the probe with the ambient helium gas. The samples and thermometers were positioned horizontally to minimize temperature gradients. In addition, and for the same purpose, samples were enclosed between two nesting copper plates (only one Cu plate is shown in Fig. 9(b)). An excitation current of about 0.1 A was run through the samples and voltage across each of them was measured to determine their respective resistance. Sample resistance R was measured as a function of sample temperature T, from room temperature T 0 down to temperatures low enough for the sample to become superconducting. RRR was defined as the ratio of R at 293 K to that at 18 K (just before the superconducting transition). The value of R(293 K) was extrapolated from R(T 0 ) by use of the empirical equation 66,67 : For HT (prior to measurements), samples were inserted individually into small-diameter alumina tubes to keep them straight.