The resistance of a circuit containing resistors in series and in parallel can be straightforwardly calculated form the resistance of each resistor - in series, the resistances are additive; in parallel, the reciprocals of the resistances (or conductances) are additive. This applies as long as the conductor dimensions are not so small as to become comparable to the electron mean free path in the material. All the components of a bronze-process Nb3Sn superconducting wire are aligned parallel to its length, and it can therefore be treated as a set of parallel conductors.
Optical microscopy image of the cross-section of a Vacuumschmelze (now European Advanced Superconductors) NS6000 superconducting wire.
The inner core is copper and the outer matrix bronze; the two are separated by a tantalum diffusion barrier. The approximately circular features are
filaments of niobium in the bronze matrix.
The resistance of each component is determined many factors. The dimensions, chemical composition (and crystallographic structure, degree of order etc.), stress and strain state, crystallographic defects (including vacancies and dislocations) and temperature all influence the resistance of a single phase. During heat treatment of the wire, many of these properties will change - for example, (trivially) the temperature, the dimensions (as niobium reacts to form the A15 phase, accompanied by a volume expansion) and the composition (as the bronze is depleted of tin). This is illustrated schematically below.
Diagram showing the set of parallel resistors describing an NS6000 superconducting wire. Before heat treatment, the A15 phase is present only as
small discontinuous regions; during heat treatment, a continuous layer forms and thickens on each niobium filament, and the bronze matrix becomes
less tin-rich.
A computer-based system has been constructed to calculate the resistivity of these superconducting wires as a function of temperature and the extent to which Nb3Sn formation has occurred. This correctly predicts the observed trends, and work is in progress to further improve the results of these calculations by measuring the resistivity (as a function of temperature) of the alloys used as wire components in practice (rather than relying on published data).
This, however, is essentially a static model - it calculates the resistance for the wire in a given condition, and that condition cannot be inferred non-destructively during measurement (but merely assessed by electron microscopy and other techniques ex situ). To avoid this limitation, the progress of the reaction between niobium and tin, and diffusion within the bronze, must be estimated as a function of time and temperature during heat treatment. To achieve this, an approximate reaction-diffusion model has been designed. Due to the appoximate circular symmetry of the wire cross-section, the most important variables - the extent of A15 phase formation and the degree of tin depletion - can be treated as functions of radial position in the wire. For efficient calculation, and to reflect this symmetry, the wire is then treated as a set of concentric annular elements (or circular elements as seen in cross-section) as illustrated below.
Schematic illustration of one of the concentric circular (annular) elements for the diffusion model.
The elements of the diffusion model can, of course, intersect the niobium filaments in various ways. The properties of each element are described by the total areas of niobium, Nb3Sn and bronze and the lengths of the interfaces (in cross-section) rather than considering each filament individually, and a finite difference method is employed to consider diffusional tin fluxes between these elements in small time intervals.
Illustration showing how one of the diffusion model elements may intersect niobium filaments.
Given the simplicity of this approach, the diffusion model produces calculated tin concentration profiles for the bronze matrix in good agreement with the measured profiles by energy dispersive x-ray spectrometry in the scanning electron microscope. These concentration variations, and the corresponding A15 layer thicknesses, can of course be used in the resistivity calculation to form a dynamic resistance model for the heat treatment of these superconductors. The predicted form of the results is in broad agreement with experiment, but these results depend very sensitively on the contributions from the alloys used in the wire - especially bronze, with a tin content varying over a very wide range. Measurements on the copper-tin alloys used in European Advanced Superconductors wires are planned to refine these calculations.
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