Significant efforts can be found throughout the literature to optimize the current-carrying capacity of Nb3Sn superconducting wires. The achievable transport current density in wires depends on the A15 composition, morphology and strain state. The A15 sections in wires contain, due to compositional inhomogeneities resulting from solid-state diffusion A15 formation reactions, a distribution of superconducting properties. The A15 grain size can be different from wire to wire, and is also not necessarily homogeneous across the A15 regions. Strain is always present in composite wires, and the strain state changes as a result of thermal contraction differences and Lorentz forces in magnet systems. To optimize the transport properties, it is thus required to identify how composition, grain size and strain state influence the superconducting properties. This is not possible accurately in inhomogeneous and spatially complex systems such as wires. This article therefore gives an overview of the available literature on simplified, well-defined (quasi-)homogeneous laboratory samples. After more than 50 years of research on superconductivity in Nb3Sn, a significant amount of results are available, but these are scattered over a multitude of publications. Two reviews exist on the basic properties of A15 materials in general, but no specific review for Nb3Sn is available. This article is intended to provide such an overview. It starts with a basic description of the niobium–tin intermetallic. After that, it maps the influence of Sn content on the electron–phonon interaction strength and on the field–temperature phase boundary. The literature on the influence of Cu, Ti and Ta additions will then be summarized briefly. This is followed by a review of the effects of grain size and strain. The article concludes with a summary of the main results.

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A Review of the Properties of Nb3Sn and Their Variation with A15 Composition, Morphology and Strain State

Sn wires and include recent findings on the variation of the upper critical field (Hc2) with temperature (T) and A15 composition. Measurements of Hc2(T) in inevitably inhomogeneous wires, as well as analysis of literature results, have shown that all available Hc2(T) data can be accurately described by a single relation from the microscopic theory. This relation also holds for inhomogeneity averaged, effective, Hc2*(T) results and can be approximated by , with t = T/Tc. Knowing Hc2*(T) implies that Jc(T) is also known. We highlight deficiencies in the Summers/Ekin relations, which are not able to account for the correct Jc(T) dependence. Available Jc(H) results indicate that the magnetic field dependence for all wires from T up to about 80% of the maximum Hc2 can be described with Kramer's flux shear model, if nonlinearities in Kramer plots when approaching the maximum Hc2 are attributed to A15 inhomogeneities. The strain () dependence is introduced through a temperature and strain dependent Hc2*(T,) and Ginzburg–Landau (GL) parameter κ1(T,) and a strain dependent critical temperature Tc(). This is more consistent than the usual Ekin unification of strain and temperature dependence, which uses two separate and different dependences on Hc2*(T) and Hc2*(). Using a correct temperature dependence and accounting for the A15 inhomogeneities leads to the remarkably simple relation , where C is a constant, s() represents the normalized strain dependence of Hc2*(0) and h = H/Hc2*(T,). Finally, a new relation for s() is proposed, which is an asymmetric version of our earlier deviatoric strain model and based on the first, second and third strain invariants. The new scaling relation solves a number of much debated issues with respect to Jc scaling in Nb3Sn and is therefore of importance to the applied community, who use scaling relations to analyse magnet performance from wire results.

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A general scaling relation for the critical current density in Nb3Sn

We review the scaling relations for the critical current density (Jc) in Nb3Sn wires and include recent findings on the variation of the upper critical field (Hc2) with temperature (T) and A15 composition. Measurements of Hc2(T) in inevitably inhomogeneous wires, as well as analysis of literature results, have shown that all available Hc2(T) data can be accurately described by a single relation from the microscopic theory. This relation also holds for inhomogeneity averaged, effective, Hc2*(T) results and can be approximated by Hc2(t)=Hc2(0) = 1-t1.52, with t = T=Tc.Knowing Hc2*(T) implies that also Jc(T) is known. We highlight deficiencies in the Summers/Ekin relations, which are not able to account for the correct Jc(T) dependence. Available Jc(H) results indicate that the magnetic field dependence for all wires from mu0H = 1 T up to about 80 percent of the maximum Hc2 can be described with Kramer's flux shear model, if non-linearities in Kramer plots when approaching the maximum Hc2 are attributed to A15 inhomogeneities. The strain (e) dependence is introduced through a temperature and strain dependent Hc2*(T,e) and Ginzburg-Landau parameter kappa1(T,e) and a strain dependent critical temperature Tc(e). This is more consistent than the usual Ekin unification of strain and temperature dependence, which uses two separate and different dependencies on Hc2*(T) and Hc2*(e). Using a correct temperature dependence and accounting for the A15 inhomogeneities leads to the remarkable simple relation Jc(H,T,e)= (C/mu0H)s(e)(1-t1.52)(1-t2)h0.5(1-h)2, where C is a constant, s(e) represents the normalized strain dependence of Hc2*(0) andh = H/Hc2*(T,e). Finally, a new relation for s(e) is proposed, which is an asymmetric version of our earlier deviatoric strain model and based on the first, second and third strain invariants. The new scaling relation solves a number of much debated issues withrespect to Jc scaling in Nb3Sn and is therefore of importance to the applied community, who use scaling relations to analyze magnet performance from wire results.

We report temperature and magnetic-field dependence of flux pinning force in bulk $\mathrm{Mg}{\mathrm{B}}_{2}$ with variable grain size. The samples are prepared by advanced methods, allowing minimizing effects of porosity, impurities, and inclusions of secondary phases. The effects of grain connectivity, flux-creep phenomena, and grain size on critical current density and flux pinning curves are analyzed. We have compared the field dependence of the pinning force for a range of samples with the predictions of theoretical models accounting for the effect of the grain size. There is qualitative agreement between grain-boundary pinning mechanism proposed by Hampshire and Jones [J. Phys. C 21, 419 (1987)] and the experimentally observed grain-size dependence of pinning force in bulk $\mathrm{Mg}{\mathrm{B}}_{2}$.

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Flux pinning force in bulk Mg B 2 with variable grain size

Bulk samples covering the entire homogeneity range of the Nb-Sn A15-phase were prepared by a new method: levitation melting under high argon pressure. The variations of the lattice parameter, superconducting transition temperature, resistivity and critical field slope were measured as a function of composition. A low-temperature X-ray diffraction study was undertaken in order to fix the compositional limit of the tetragonal phase. The theoretical expectations for the critical field slopes at the transition temperature, Tc, based on actually-observed alloy parameters, were found to be in good agreeement with measured values.

The physical and structural properties of superconducting A15-type Nb-Sn alloys

The critical current Ic as a function of applied tensile strain depends upon the residual stress state that exists in Nb3Sn bronze‐process composite conductors after cooling from the reaction temperature to 4.2 K. This paper presents a simple analytical method to predict the residual stress state and the stress‐strain curves of Nb3Sn composites at 4.2 K. The method requires only a prior knowledge of the volume fraction of the composite components and the percentage of Nb converted to Nb3Sn. Experimental results and analytically derived values are compared and good agreement is generally found.

A prediction of the stress state in Nb3Sn superconducting composites

The semiempirical scaling law for flux pinning, Fp=ABnc2bl (1−b)m has been tested for bronze process Nb3Sn conductors under uniaxial tension. The dependences of the bulk pinning force Fp on the upper critical field Bc2 and the reduced field, b=B/Bc2, were weakly affected by strain. However, the factor A, which depends on the number and strength of the pinning centers, and on the Ginzburg‐Landau parameter κ decreased by more than a factor of 20 when the compressive prestrain on the Nb3Sn was removed by application of external stress. The variations of Tc and Bc2 with strain suggest that the change in κ is not sufficient to account for the change in A. Therefore, it is probable that strain induces microstructural changes which affect the number and/or strength of the pinning centers.

The effect of strain upon the scaling law for flux pinning in bronze process Nb3Sn

A model is presented of the simultaneous spreading and cooling of the liquid drop in a hammer and anvil apparatus for rapid quenching of liquid metals. The viscosity of the melt is permitted to vary with temperature, and to avoid mathematical complications which would be associated with spatial variation of the viscosity, Newtonian cooling is assumed. From an expression for the force required to spread the specimen, coupled equations for the mechanical energy balance for the system and the heat transfer from the sample to the hearth and hammer were obtained, and solved numerically. The sample reaches its final thickness when the force required to deform it becomes greater than the force exerted on it by the decelerating hammer. The model was fit to measurements of sample thickness versus hammer speed, using the interface heat transfer coefficient, h, as an adjustable parameter. The values of h so obtained vary somewhat with the melt alloy/substrate metal combination. From predicted cooling curves, the effects of hammer speed, sample size, and initial melt temperature on the cooling rate and the efficiency of glass formation can be assessed. Addition of sample superheat shifts the cooling curve relative to the expected position of the time‐temperature‐transformation curve for formation of crystalline material from the melt, and thus is an effective means of increasing the probability of glass formation in this type of apparatus.

D. S. Easton

Papers

A prediction of the stress state in Nb3Sn superconducting composites

The effect of strain upon the scaling law for flux pinning in bronze process Nb3Sn

A study of cooling rates during metallic glass formation in a hammer and anvil apparatus