^{1}, C. M. Fischer

^{2}, P. J. Lee

^{2}and D. C. Larbalestier

^{2}

### Abstract

In powder-in-tube composites, the phase forms between a central tin-rich core and a coaxial tube, thus causing the tin content and superconducting properties to vary with radius across the layer. Since this geometry is also ideal for magnetic characterization of the superconducting properties with the field parallel to the tube axis, a system of concentric shells with varying tin content was used to simulate the superconducting properties, the overall severity of the composition gradient being defined by an index . Using well-known scaling relationships and property trends developed in an earlier experimental study, the critical current density for each shell was calculated, and from this the magnetic moment of each shell was found. By summing these moments, experimentally measured properties such as pinning-force curves and Kramer plots could be simulated. We found that different tin profiles have only a minor effect on the shape of Kramer plots, but a pronounced effect on the irreversibility fields defined by the extrapolation of Kramer plots. In fact, these extrapolated values are very close to a weighted average of the superconducting properties across the layer for all . The difference between and the upper critical field commonly seen in experiments is a direct consequence of the different ways measurements probe the simulated gradients. gradients were found to be significantly deleterious to the critical current density , since reductions to both the elementary pinning force and the flux pinning scaling field compound the reduction in . The simulations show that significant gains in of strands might be realized by circumventing strong compositional gradients of tin.

This work was supported by the US Department of Energy, Office of High Energy Physics. The facilities used for the work were partially supported by the NSF MRSEC for Nanostructured Materials and Interfaces at the University of Wisconsin.

I. INTRODUCTION

II. SIMULATIONS

III. DISCUSSION

A. Effect of tin gradients on the apparent irreversibility field

B. Effects on flux pinning and the critical current density

C. Features of real tin composition profiles

IV. CONCLUSIONS

### Key Topics

- Tin
- 48.0
- Niobium
- 41.0
- Superconductivity
- 29.0
- Conductors
- 15.0
- Critical currents
- 11.0

## Figures

Top: Scanning electron microscope image of a cross-section of a composite. Bottom: Schematic of a typical configuration for electromagnetic measurements, showing the arrangement of magnetization current shells with respect to the applied field and the radial direction of tin diffusion.

Top: Scanning electron microscope image of a cross-section of a composite. Bottom: Schematic of a typical configuration for electromagnetic measurements, showing the arrangement of magnetization current shells with respect to the applied field and the radial direction of tin diffusion.

Input profiles of tin composition as a function of radius. The index depicts the sharpness of the profile, where , and are shown here and in Figs. 4 to 7. An ideal profile, in which the tin content is constant at , is also shown. Local values of the critical temperature and upper critical field (at ) that correspond to the local composition are indicated on the right axes.

Input profiles of tin composition as a function of radius. The index depicts the sharpness of the profile, where , and are shown here and in Figs. 4 to 7. An ideal profile, in which the tin content is constant at , is also shown. Local values of the critical temperature and upper critical field (at ) that correspond to the local composition are indicated on the right axes.

Maximum bulk pinning force plotted against the irreversibility field, derived by extrapolating Kramer plots, for a series of samples, measured at . Despite differences in heat treatment duration ( at ) and temperature, these data collapse onto the single linear fit given by Eq. (5). Data taken at to (open symbols) yield very low values of , but are still consistent with those at lower temperatures used for Eq. (5).

Maximum bulk pinning force plotted against the irreversibility field, derived by extrapolating Kramer plots, for a series of samples, measured at . Despite differences in heat treatment duration ( at ) and temperature, these data collapse onto the single linear fit given by Eq. (5). Data taken at to (open symbols) yield very low values of , but are still consistent with those at lower temperatures used for Eq. (5).

Moment of the shells as a function of radius simulated at and (a), and at and (b). The indices for the curves are the same as for Fig. 2. Note that the coordinate has been mapped to a radius to simulate actual composites.

Moment of the shells as a function of radius simulated at and (a), and at and (b). The indices for the curves are the same as for Fig. 2. Note that the coordinate has been mapped to a radius to simulate actual composites.

Critical current density, as would be determined from a magnetization experiment of an actual strand, as a function of field simulated at (a) , and (b) . The inset of plot (a) shows a magnified view of the mid field data. The dashed and dash-dot curves correspond to the noncopper values determined by magnetization measurements in Ref. 17 for ternary and binary strands with the field parallel to the strand axis (as in Fig. 1). The large diamonds correspond to transport measurements for the same ternary strands reported in Ref. 32. The indices for the curves follow the same sequence as in Fig. 2.

Critical current density, as would be determined from a magnetization experiment of an actual strand, as a function of field simulated at (a) , and (b) . The inset of plot (a) shows a magnified view of the mid field data. The dashed and dash-dot curves correspond to the noncopper values determined by magnetization measurements in Ref. 17 for ternary and binary strands with the field parallel to the strand axis (as in Fig. 1). The large diamonds correspond to transport measurements for the same ternary strands reported in Ref. 32. The indices for the curves follow the same sequence as in Fig. 2.

Bulk flux-pinning force curves as a function of field simulated at (a) and (b) . The indices for the curves follow the same sequence as in Fig. 2.

Bulk flux-pinning force curves as a function of field simulated at (a) and (b) . The indices for the curves follow the same sequence as in Fig. 2.

Kramer function curves as a function of field simulated at (a) and (b) . The indices for the curves follow the same sequence as in Fig. 2. The insets in both plots show a magnified view of the region near the field axis.

Kramer function curves as a function of field simulated at (a) and (b) . The indices for the curves follow the same sequence as in Fig. 2. The insets in both plots show a magnified view of the region near the field axis.

Energy-dispersive x-ray spectroscopy analyses of a strand in Ref. 14 are simulated using the tin composition profile shown in plot (a), with corresponding local values of critical temperature and upper critical field indicated on the right axes. The open boxes denote data points in Ref. 14 for a , reaction. In plot (b), the simulated critical current density vs field curve at is shown over a field range of technological interest. Large diamonds denote measured values in Ref. 13 for similar strands as in Ref. 14. In plot (c), the corresponding bulk pinning force at is presented. Plot (d) shows the Kramer function, along with its extrapolation (dashed line) from data at and below. In all plots, the curve corresponding to an ideal profile of a constant is also labeled.

Energy-dispersive x-ray spectroscopy analyses of a strand in Ref. 14 are simulated using the tin composition profile shown in plot (a), with corresponding local values of critical temperature and upper critical field indicated on the right axes. The open boxes denote data points in Ref. 14 for a , reaction. In plot (b), the simulated critical current density vs field curve at is shown over a field range of technological interest. Large diamonds denote measured values in Ref. 13 for similar strands as in Ref. 14. In plot (c), the corresponding bulk pinning force at is presented. Plot (d) shows the Kramer function, along with its extrapolation (dashed line) from data at and below. In all plots, the curve corresponding to an ideal profile of a constant is also labeled.

## Tables

Comparison of the weighted mean of the upper critical field with extrapolations of simulated Kramer functions, at and .

Comparison of the weighted mean of the upper critical field with extrapolations of simulated Kramer functions, at and .

Variation of flux-pinning quantities with tin profile index.

Variation of flux-pinning quantities with tin profile index.

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