Invariant formulation of the strain dependence of the critical temperature Tc of Nb3Sn in a three term approximation

It has been shown previously in a full detailed analysis that the strain dependence of the critical temperature may be obtained from a general strain invariant formulation of T_c in strong superconductivity. A physical model was presented in which the phonon frequency spectrum is represented through generalized elastic stiffness coefficients that include strain dependence. The primary purpose of the present work is to achieve a simplification of the analysis in order to facilitate calculation and reveal the essential physical content. The formulation in wave vector space of the equations for T_c in strong superconductivity is reviewed. The method of simplification employs a succession of approximations to the effective elastic constants that enter the relation between phonon frequency and wave number. It is found that the effective elastic constants in the crystal symmetry directions may be grouped into sets having similar form, and this form includes terms in common among the sets and difference terms. The difference terms are found to be in the nature of gradients and may be eliminated to good approximation. The common terms include the strain dependence in a form identified as a deformation strain parameter. The analysis treats spherical (hydrostatic) and deformation strain dependence under longitudinal and transverse applied strain for wire and tape conductor. The analysis is applicable over a full range of applied strain, including small strains often described by a power law strain dependence, and larger strains often described by a deviatoric strain approach. A comparison is provided between the results of the full detailed analysis and the results of the approximate treatment showing the degree of agreement in the various applied strain orientations.     

Invariant temperature and field strain functions for Nb3Sn composite superconductors

The observed strain dependence of Nb₃Sn over an extended range of applied strain requires revision of the functions used previously to represent that strain dependence. The concept and various definitions of the strain function for the upper critical field and the critical temperature are reviewed. An invariant strain function for the critical temperature is derived from the strain energy potential of the general invariant strain analysis. The invariant strain function is applied to a study of uniaxial longitudinal applied strain on wire and tape conductors. Uniaxial forms of the general strain function are derived, and parameter values of the strain function are determined from a fit of the uniaxial forms to published uniaxial strain data for the critical temperature. The concept of precompression is examined in the context of the strain function. The invariant strain function, as an analytic function, is compared with numerical calculations of the general invariant strain analysis for longitudinal and transverse applied strain. The correspondence between the strain dependence of the critical temperature and the upper critical field is discussed, and a form of the invariant strain function for the upper critical field is presented.     

The study on the critical temperature, composition, and microstresses in a Nb3Sn layer

Monofilamentary Nb₃Sn wires of large diameter with niobium tube, which were obtained by the method of solid-phase diffusion, are well suited for the study of the distribution of the critical temperature T_c in Nb₃Sn layers. Three regions with different gradients of Sn and Nb concentration and different Cu content can be distinguished in Nb₃Sn layer. In the central part of the layer, the Sn content comprises 24.5 at.% and the gradient of Sn concentration is negligibly small. Measurements on specimens of 1.2 mm in diameter with a slit cut along the cylinder generatrix showed that the critical temperature of the Nb₃Sn region adjacent to Cu(Sn) bronze is lower than the critical temperature of the central part of the layer. Fluctuations of T_c in the central part of the layer exceed the change of T_c related to the gradient of the Sn concentration, which is very small. These fluctuations spread both the R(T) curve and the high-temperature part of the temperature transition registered by the inductive method.     

Elastic stiffness model for the critical temperature Tc of Nb3Sn including strain dependence

A calculation is established for the critical temperature T_c of the superconductor Nb₃Sn that includes the dependence on applied mechanical strain. The calculation employs the formalism of strong coupling phonon superconductivity, as usually given in frequency space. The directional nature of strain is included by expressing the equations of strong coupling in wave vector space. The relation between wave number and frequency is provided by the dispersion relations incorporating the effective elastic constants for the symmetry directions of the cubic crystal. An analytical formalism is established in which the elastic constants are derived in a unified way from an assumed strain energy potential function. The form of the strain energy potential is governed by the cubic symmetry. The scalar invariants of the strain tensor under the cubic symmetry group are determined as a set of basis functions for the strain energy potential. In the harmonic approximation, the relation between the strain energy function and the elastic constants determines the harmonic amplitudes of the strain potential from the measured dispersion relations. The electron-phonon coupling characteristic is approximated in a simple analytic form determined by inspection of the experimentally determined tunneling and phonon density of states. The critical temperature is calculated, through the equations of strong coupling, as a sum over the crystal symmetry directions. The anharmonic terms of the strain energy potential are introduced as the source of the strain dependence of the critical temperature. The allowed form of the anharmonic terms is again governed by cubic symmetry. The amplitudes of the anharmonic terms are determined from the strain dependence characteristics of single crystals and composite superconductors. The calculations are found to represent the observed strain dependence well with a strain energy function that contains three scalar invariants of the strain tensor, including the spherical (hydrostatic) strain invariant, and the principle parts of the second and third invariants of the deviatoric strain tensor. The formalism is applied to the analysis of composite conductors. The characteristics of the strain dependence of wire and tape geometries under longitudinal and transverse loads are related to the symmetry of the conductor and direction of applied load. Implications of conductor symmetry and constraint on the measurement of the strain dependent properties are identified.     

Enhancement of critical current density of YBa2Cu3O7 − δ thin films by self-assembly of Y2O3 nanoparticulates

YBa₂Cu₃O_7 − δ (YBCO) thin films, possessing high critical current density (J_c), have been synthesized by embedding a homogeneous array of Y₂O₃ non-superconducting nanoclusters/nanoparticles using a pulsed laser deposition technique. The size, interparticle spacing, and density of Y₂O₃ nanoparticles in YBCO thin films were tailored by varying the number of laser pulses in order to determine the optimum size for effective immobilization of vortices. Scanning transmission electron microscopy with atomic number contrast and X-ray diffraction techniques were used to determine the size and structure of the nanoparticles. Both techniques indicate that the Y₂O₃ particles are epitaxial with respect to the surrounding YBCO matrix. The information about pinning of vortices by the nanoparticles was obtained by investigating the behavior of critical current density as a function of temperature and applied field, which in turn determines the vortex density in the sample. The superconducting transition temperature (T_c) of YBCO films with the inclusion of nanoparticles was observed to remain almost the same or decrease marginally (1-2 K) with respect to T_c of pure YBCO films deposited under identical conditions. However, J_cs of YBCO films embedded with self-assembled nanoparticles were found to be significantly higher than that of pure YBCO films. The J_c enhancement was up to five times in high magnetic field, which is a key requirement for practical application of high-T_c materials.     

Percolation and the resistive transition of the critical temperature Tc of Nb3Sn

The relationship between the distribution of the critical temperature, the percolation function, and the resistive transition of the critical temperature is explored for polycrystalline Nb₃Sn. In the neighborhood of the critical temperature, Nb₃Sn is assumed to be a random mixture of superconducting and normal grains. Percolation concepts are applied to a study of the resistivity. A general analysis is made showing that the onset and shape of the resistive transition for composite conductors are determined by the percolation function and the distribution of the critical temperature. An approximate form of the percolation function is determined based on a linear FEM analysis. Example resistive transitions are calculated for an assumed normal distribution of the critical temperature. An argument is presented that relates grain orientation and strain dependence in Nb₃Sn. It is noted that a dependence of the distribution of T_c with strain, in addition to the usual shift in T_c with strain, would be the result of a strain dependence that is a function of grain orientation. The analysis shows the extent to which the slope of the resistive transition is a measure of the distribution of the critical temperature, and therefore a measure of the grain orientation strain sensitivity. Finally, a method is described to determine the percolation function experimentally.     

Critical current of react-and-jacket processed Nb3Sn conductor

A new type of large-scale Nb₃Sn conductor was developed that has an aluminum-alloy jacket to support an electromagnetic force. The manufacturing process of the conductor has a unique feature, in which the jacketing process is performed after a reaction heat treatment of the Nb₃Sn cable. This enables the conductor to have a high critical current, because the thermal strain of the Nb₃Sn filaments is decreased. Critical current measurements using a short conductor sample confirmed the expected high performance.     

Magnetic field orientation dependence of critical current in industrial Nb3Sn strands

In usual superconducting devices such as magnets for NMR, the magnetic field is perpendicular to the superconducting strand axis. But in some special devices, such as magnets for the toroidal field system of fusion machines, the strands can experience any field orientation. For NbTi strands, the pinning force is dependent on the field orientation because of the drawing process (Takacs, S., Polak, M. and Krempasky, L., Critical currents of NbTi tapes with differently oriented anisotropic defects, Cryogenics, 1983, 23, 153–159). In the case of Nb₃Sn strands, the draw and react process suggests that the pinning force is isotropic. In fact, preliminary experiments have shown the contrary, which is why the magnetic field orientation dependence of the critical current for two types of industrial Nb₃Sn strands has been measured. These measurements have been performed for seven field orientations at field strengths up to 20 T. A clear anisotropic effect has been observed, which cannot be explained by Kramer's pinning law. The results are in very good agreement with an empirical law proposed in a recent study by Takayasu et al. (Takayasu, M., Montgomery, D.B. and Minervini, J.V., Effect of magnetic field direction on the critical current of twisted multifilamentary superconducting wires, Inst. of Phys. Conf. Ser., 1997, 158, 917–920). The parameters to be used in this law could be specific to the manufacturing process.     

Finite element simulations of elasto-plastic processes in Nb3Sn strands

The manufacturing of Nb₃Sn strands, with drawing and annealing of multifilamentary strands followed by a heat treatment at about 900 K to form the Nb₃Sn by reaction of tin and niobium, has the potential to create a complex internal stress system. The strain sensitivity of the Nb₃Sn superconducting properties makes prediction of the internal stresses a necessary step to understanding the performance of Nb₃Sn conductors under the magnetic load conditions experienced in a coil. An elasto-plastic one dimensional finite element model, including temperature dependent stress-strain curves, annealing and manufacturing process stresses, is used to derive the internal stresses of Nb₃Sn strands. The model is benchmarked against a range of experimental data, including stress-strain tensile tests, superconducting critical current-strain tests, and length changes through heat treatment and through a 4 K thermal cycle. The model can predict all the experimental features and shows a number of unexpected conclusions regarding the origin of the Nb₃Sn stresses.     

High-field magnetic and transport critical current density of Sm-Ba-Cu-O: Agx superconductors at low temperature

Variation of magnetic critical current density (J _mc) and transport critical current density (J _tc) in high magnetic fields at liquid helium temperature were investigated on Sm-Ba-Cu-O: Ag _x (x=0.0, 0.2, 0.4, and 0.6) superconducting samples prepared by the coprecipitation technique.T _c and crystallographic parameters remain unaffected by Ag addition. However, bothJ _mc andJ _tc were found to increase on Ag addition. The volume pinning force (F _p) also increased with increase in Ag concentration, resulting in stronger flux line sheer. Microwave-induced d.c. voltage measurement shows a significant reduction of the total number of weak links between the superconducting grains with increasing concentration of Ag, which creates stronger pinning potentials between the boundary of the superconducting intergrains. Hence, the volume pinning force is greatly enhanced by Ag addition, leading to flux line lattice movement which is responsible for highJ _mc andJ _tc.     

Magnetic flux pinning and flux jumps in polycrystalline MgB2

MgB₂ polycrystalline samples were fabricated under varying conditions of isostatic pressing in argon gas. The critical current densities (J_C) were determined through measurements of hysteresis loops, and the highest value of J_C at 10 K was 1.9 × 10⁴ A/cm² at 4.8 T. The depinning temperatures were measured at various magnetic fields using the vibrating reed technique. Flux jumps appeared below 7.4 K. The hysteresis loops were carefully examined to determine the temperature and magnetic field range where flux jumps appeared.     

DC electrical resistivity and magnetic studies in Yttrium Barium Copper oxide/barium titanate composite thin films

YBCO + BaTiO₃ composite thin film is synthesized by pulsed laser deposition. Fluctuations on the electrical conductivity were investigated for zero fields. The logarithmic plots of excess conductivity and reduced temperature reveals two distinct regions namely mean field region and short wave fluctuation region. Dimensionality crossover occurs from 3D to 2D at temperature above the transition temperature. The contribution of weak link effect is calculated. The phase formation and grain alignments were analyzed by X-ray diffraction and scanning electron microscopy techniques. Enhancement of flux pinning increases the critical current density in the composite and develops strong pinning force in the material.     

Field and temperature dependencies of critical current on industrial Nb3Sn strands

The increasing need for high field magnetic devices has focused attention on filamentary Nb₃Sn conductors, whose critical data are superior to NbTi conductors. To choose the suitable operating parameters and to determine the stability margin of magnet systems, it is very important to know the effect of temperature and magnetic field on the superconducting properties, especially on the critical current. Up to now, for design calculation, the so-called “Summers model” was assessed theoretically on experimental data obtained by Spencer et al., (The temperature and magnetic field dependence of superconducting critical current densities of multiinflammatory Nb₃Sn and NbTi composite wires. IEEE Trans Mag, Mag-15 (1979) 76) and Suenaga et al., Superconducting critical-current densities of commercial multifilamentary Nb₃Sn(Ti) wires made by the bronze process. Cryogenics (1985) 25, 123). Apart these very useful preliminary experimental data, very little has been done on the very different industrial strands which are now produced in the industry. Industrial Nb₃Sn strands are generally tested and checked only at 4.2 K and their operating design temperature is often very different, sometimes around 6 K. It is now urgent to validate the model and to confirm that the data taken up to now in the design calculations are conservative.     

Forced flow He vapor cooled critical current testing facility for measurements of superconductors in a wide temperature and magnetic field range

As superconducting materials find their way into applications, there is increasing need to verify their performance at operating conditions. Testing of critical current with respect to temperature and magnetic field is of particular importance. However, testing facilities covering a range of temperatures and magnetic fields can be costly, especially when considering the cooling power required in the cryogenic system in the temperature range below 65 K (inaccessible for LN₂). Critical currents in excess of 500 A are common for commercial samples, making the testing of such samples difficult in setups cooled via a cryocooler, moreover it often does not represent the actual cooling conditions that the sample will experience in service. This work reports the design and operation of a low-cost critical current testing facility, capable of testing samples in a temperature range of 10–65 K, with magnetic field up to 1.6 T and measuring critical currents up to 900 A with variable cooling power.     

5CrMnMo应变率和温度相关力学性能实验研究

在较大的温度(25℃-537℃)和应变率(10-4s-1-10-2s-1)范围内对5CrMnMo进行了拉伸实验,获得了相应的应力应变曲线.试验结果表明在室温和试验的应变率范围内(10-4s-1-10-2s-1),5CrMnMo的力学性能是应变率无关的.随着温度的升高,出现了模量E、屈服强度σs和抗拉强度σb的应变率强化效应和温度弱化效应;还出现了加工硬化倾向减小的机制和蠕变效应增大机制,且温度越高这两种机制越强,应变率越高这两种机制越弱.在这两种机制作用下,温度越高失稳应变εb越小,断后伸长率δ50越大;但应变率越高δ50越小.当试验温度较高且应变率较低时,伴随有马氏体板条向拉伸方向偏转的细观特征.     

Addition of Ho2O3 of different types to MgB2 in the ex-situ Spark Plasma Sintering: Simultaneous control of the critical current density at low and high magnetic fields

Two different types of Ho₂O₃ powders (showing a much different morphology) were added to MgB₂ in the ex-situ Spark Plasma Sintering (SPS). In the 5–25 K range, the first Ho₂O₃ powder type does not significantly suppress the critical current density J_c at low magnetic fields and the second one enhances it at high fields, while their mixture simultaneously controls J_c at both small and high magnetic fields so that the decrease is small at low fields and there is a notable enhancement at high fields when compared to pristine sample. The control of J_c(H) is discussed versus specific characteristics of the raw powders, the resulting microstructure of the added SPS-ed samples and pinning details from magnetic relaxation measurements.     

Critical currents density and current loops range in MgB2 thin layers obtained by the technique of ions implantation followed by pulsed plasma transient annealing

We report on the measurements of transport and magnetic properties of MgB₂ thin films obtained using ion implantation technique followed by irradiation with high intensity pulsed plasma beams. To get detailed information about the process of current percolation in a superconducting phase, about current transport and size of the current loop, we used the method of reverse leg of magnetisation loop. On the basis of this method we have found that the technique of implantation allowed one to obtain a continuous layer of MgB₂ superconductor characterized by the critical current density of J_c = 1.25 × 10⁵ A/cm².     

Stress/strain characteristics of Cu-alloy sheath MgB2 superconducting wires

The mechanical properties of Cu and Cu-alloy (Cu-Zr, Cu-Be and Cu-Cr) sheath in situ PIT-processed MgB₂ superconducting wires were studied at room temperature (RT) and 4.2 K. The effects of stress/strain on the critical current (I_c) of the wires have also been studied at 4.2 K and in magnetic fields up to 5 T. Alloying the Cu sheath significantly increased the yield stress of the wires. The 0.5% flow stresses of the Cu-alloy sheath wires were 147-237 MPa, whereas that of Cu was 55 MPa. At RT, the serration in the stress-strain curves corresponding to the multiple cracking was observed around a strain of 0.4% and the curve almost saturated beyond that point. The strain dependence of I_c prior to the critical strain (ε_irr) was different depending on the magnetic field; being almost constant at 2 T and increased with strain at 5 T. The I_c decreased beyond ε_irr, which was much larger for Cu-alloy sheath wires as compared with Cu sheath wire. The magnitude of ε_irr is due to the difference in the thermal compressive strain in the MgB₂ core, which was relaxed by yielding in the sheath materials. The transverse compression tests revealed that the I_c of the Cu-alloy sheath wire did not degrade up to about 95 MPa, which is also higher than that of Cu sheath wire.     

Testing field and annealing temperature dependence of leakage properties in Bi3.25La0.75Ti3O12 thin films

Ferroelectric Bi_3.25La_0.75Ti₃O₁₂ thin films annealed at different temperatures were prepared on Pt/Ti/SiO₂/Si substrates by metalorganic decomposition method. The leakage current behavior and the current conduction mechanism were investigated. For all films, the leakage current density − electric field (J−E) characteristic is confined within a “triangle” in the log (J)− log (E) plane bounded by three limiting curves: Ohm's law (J ∝ E), trap-filled-limit (J ∝ E^a, a > 1), and Child's law (J ∝ E²). At high field region, Bi_3.25La_0.75Ti₃O₁₂ thin films with grains of rod-like show higher leakage current, while films with grains of spherical- or planar-like exhibit lower leakage current.     

Multiscale analysis of the influence of the triplet helicoidal geometry on the strain state of a Nb3Sn based strand for ITER coils

A theoretical model of a beam of unidirectional composites—based on the homogenisation theory and a refined kinematical hypothesis—is used for the analysis of the influence of the helicoidal geometry of a superconducting strand triplet on its strain state. The triplet is the first cabling stage of the superconducting cables used to wind the coils of ITER fusion reactor. The multiscale modelling strategy is presented, for which a finite element code has been developed. A triplet of Nb₃Sn based strands subjected to an axial stretch is analysed, and the resulting complete 3D strain state in the Nb₃Sn filament is recovered. An “extra” strain is found due to the helicoidal geometry of the triplet. Discussion of the results concludes the paper.