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.     

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.     

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.     

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.     

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.     

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.     

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.     

Quench analysis of an ITER 13T-40kA Nb3Sn coil (CS insert)

In order to analyze the quench characteristic of a cable-in-conduit (CIC) conductor that has a sub-cooling channel at the center of conductor cross section, an axisymmetric two-dimensional calculation model was developed. The tests and calculation results of the central solenoid (CS) insert were compared regarding the pressure drop and the behavior of the total voltage, temperature and normal zone propagation in the quench. They show good agreement. Therefore, the effectiveness of the calculation model is verified. It was also found that there is coolant convection between the central channel and bundle region even in a steady state. This makes the pressure drop in the central channel larger than that in a cylindrical pipe which has a smooth surface. In addition, it was found that the higher temperature of the coolant flowing through the central channel heats the coolant and the cable in the bundle region. It can be said that the hot coolant flowing through the central channel accelerates normal zone propagation.     

Influence of a degraded SrTiO3 layer at the YBa2Cu3O7−δSrTiO3 interface on the dielectric behavior at cryogenic temperature

In an YBa₂Cu₃O_7−δ/SrTiO₃/YBa₂Cu₃O_7−δ parallel capacitor fabricated by a chemical mechanical planarization method, the dielectric constant and loss in the presence of electric fields at 2.2 K were more than 26,000 and less than 0.027, respectively. We propose a multilayer model that explains this behavior. The model assumes that the SrTiO₃ film is composed of single-crystal-like SrTiO₃ layers with a dielectric constant ε_r = 30,000 and degraded SrTiO₃ layers with dielectric constants that vary continuously from 25, the dielectric constant of YBa₂Cu₃O_7−δ, to 30,000. Results of a numerical calculation revealed that the thickness of the single-crystal-like SrTiO₃ layer was more than 92% of a 600-nm-thick SrTiO₃ film.     

Spatial distribution of the local resistive transition and the critical current density in YBCO coated conductors using Low-temperature Scanning Laser Microscopy

Low-temperature Scanning Laser Microscopy (LTSLM) was carried out to reveal the spatial distribution of the local resistive transition temperature and the local current density in commercial YBCO coated conductors near the superconducting transition. The result of the sample with an array of holes of various sizes shows that the signal δV is proportional to the current density. The distribution of the critical temperature of the sample with two parallel bridges is quite homogeneous and the transport current flows mainly along the outer edge of the sample. Using LTSLM we directly image the current path in YBCO coated conductors of different patterns.     

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.     

Projected upper limits to the critical current density of superconducting Nb3Ge, Nb3Sn, and V3Ga

Applying a phenomenological theory of flux pinning developed by Kramer,¹ where the ultimate critical current density (J_c) of a superconductor is determined by plastic shearing of the flux lattice, approximate upper limits are put on J_c for Nb₃Ge, Nb₃Sn, and V₃Ga. At 4.2 K and for magnetic fields H < 100 kG, the J_c of V₃Ga is greater than that of either Nb₃Ge or Nb₃Sn and Nb₃Sn has somewhat higher values than Nb₃Ge. Above 200 kG Nb₃Ge has the highest J_c due to its having the highest upper critical field and at 14 K or above it probably has the largest J_c at all field values.     

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.     

Influence of the common varistor dopants (CoO, Cr2O3 and Nb2O5) on the structural properties of SnO2 ceramics

The influence of dopants commonly used in SnO₂ varistor ceramics, such as CoO, Cr₂O₃ or Nb₂O₅, on the structural properties of SnO₂ was investigated. Several SnO₂-based ceramics containing only one of the dopants were prepared and characterized. Spectroscopic investigations [visible, near infrared (IR) and IR region] were performed to obtain information about dopants valence states inside the ceramics, as well as about their influence on electronic structure of the material. Structural properties were investigated by X-ray diffraction analysis and mechanisms of dopant incorporation were proposed. Obtained results were confirmed with results of the electrical measurements. Microstructural changes in doped ceramics were investigated by scanning electron microscopy (SEM) analysis that showed great differences in densities, grain size, and morphology of the SnO₂ ceramics depending on type of dopants and their distribution.     

A study on the phase transition and characteristics of rare earth elements doped BaTiO3

Non-reducible dielectric composition satisfying X8R specifications (−55 to 150°C, ±15%) for automotive applications was investigated in a BaTiO₃-MgCO₃-MnO₂-rare earth elements (La, Pr, Gd, Tb, Dy, Ho, Er, Yb, Lu) system. In the cases of Er, Yb, Lu-substituted samples, a transition of the Curie point shifted to a higher temperature was observed. The peak shift toward higher temperatures indicated that the tetragonal phase became more stable at the high temperature. The ionic radius of rare earth elements was found to be an important factor in controlling the temperature dependency of the dielectric properties. Especially, smaller ionic radii of rare earth elements such as Yb and Lu were effective dopants to meet the X8R temperature-capacitance characteristics.     

Preparation and characterization of porous Nb2O5 nanoparticles

Porous niobium oxide (Nb₂O₅) nanoparticles have been successfully prepared without any surfactant assisting. They were characterized by X-ray diffraction (XRD), nitrogen sorption, transmission electron microscopy (TEM) and scanning electron microscopy (SEM). The results show that the porous Nb₂O₅ nanoparticles are polycrystalline, the Brunauer–Emmett–Teller (BET) surface area is 12.09 m²/g and the average pore size is 3.4 nm. In addition, spherical and flake-like Nb₂O₅ samples were obtained and characterized. Possible explanations for the formation of Nb₂O₅ nanocrystals with different morphologies are discussed.     

Stabilization of the cubic A-15 phase of Nb3Sn

From ac heat capacity measurements, it has been found that copper-plating small Nb₃Sn samples appreciably inhibits the cubic-tetragonal transformation.     

TG/DSC analysis of Fe8(OOH)16Cl1.3 nanospindles

The thermal analysis of Fe₈(OOH)₁₆Cl_1.3 (Akaganeite-M) nanospindles prepared by the hydrolysis of FeCl₃ solutions are determined by thermogravimetric analyses and differential scanning calorimetry (TG/DSC), in conjunction with field-emission scanning electron microscopy (FE-SEM), and X-ray diffraction (XRD). Different products are formed after Fe₈(OOH)₁₆Cl_1.3 nanospindles are calcined at different temperatures for 30 min in N₂ atmosphere: Fe_1.833(OH)_0.5O_2.5 and magnetite obtained at 250 °C; pure magnetite (Fe₃O₄) obtained at 630 °C; and magnetite containing some iron nitrides (Fe₂N and Fe₄N) obtained at 800 °C. The calcination of Fe₈(OOH)₁₆Cl_1.3 provides a new method to prepare pure magnetite.