Flux Pinning Analysis
Theory: the force that holds flux still
Overview
Why does a superconducting wire carry less current in stronger fields?
In practical superconductors, field enters as thin quantised flux lines. Current exerts a Lorentz force J×B pushing them along; if they move, a voltage appears and energy is lost, so the material is no longer superconducting in effect. Tiny precipitates and grain boundaries catch the flux lines and hold them (pinning). Critical current density is where Lorentz force balances pinning force. As field rises the flux lines multiply, pinning per line weakens, and near the upper critical field superconductivity itself fades, so allowable current drops sharply.
Pinning-force scaling law
NbTi (4.2 K): Bc2 = 10.5 T, p = 1, q = 1, Jc(5 T) = 3×10⁹ A/m². Nb3Sn (4.2 K): Bc2 = 27 T, p = 0.5, q = 2, Jc(12 T) = 2.5×10⁹ A/m² (non-copper area). C is fixed from that single point.
Flux-line spacing and force per line
a is the spacing of flux lines in a triangular lattice; Φ₀ is the flux quantum.
Tiny titanium layers inside NbTi
NbTi wire's high critical current density comes from pinning by fine α-titanium layers precipitated during heat treatment. Repeated drawing flattens the precipitates into ribbons a few nanometres thick; pinning is most effective when their spacing approaches the flux-line spacing of tens of nanometres. Optimising this combination of drawing and heat treatment in the 1980s greatly raised NbTi's critical current density and underpinned the spread of MRI.
Worked examples
Example 1: NbTi (4.2 K) critical current density
| Field | Jc (A/mm²) | Fp (N/m³) |
|---|---|---|
| 2 T | 4,636 | 9.3×10⁹ |
| 5 T | 3,000 | 1.50×10¹⁰ |
| 7 T | 1,909 | 1.34×10¹⁰ |
| 9 T | 818 | 7.4×10⁹ |
| 10 T | 273 | 2.7×10⁹ |
Example 2: Nb3Sn (4.2 K) critical current density
| Field | Jc (A/mm²) | Fp (N/m³) |
|---|---|---|
| 8 T | 4,913 | 3.9×10¹⁰ |
| 12 T | 2,500 | 3.0×10¹⁰ |
| 16 T | 1,164 | 1.9×10¹⁰ |
| 20 T | 422 | 8.4×10⁹ |
| 24 T | 71 | 1.7×10⁹ |
Fp peaks at 5.2 T for NbTi (b = 0.5) and 5.4 T for Nb3Sn (b = 0.2). Flux-line spacing is 49 nm at 1 T, 22 nm at 5 T and 14 nm at 12 T. At 3×10⁹ A/m² the force on each flux line is 6.2×10⁻⁶ N per metre of length.
At 10 T NbTi is nearly useless, but Nb3Sn still has margin.
NbTi loses superconductivity at 10.5 T, so its Jc plunges just below that. Nb3Sn has a high upper critical field of about 27 T, and since grain boundaries dominate its pinning, its peak sits at lower reduced field and Jc declines gently at high field. Hence Nb3Sn for research magnets above 15 T and fusion magnets. But Nb3Sn is brittle and its Jc falls with strain, so manufacture is harder — for instance heat-treating after winding. For still higher fields, high-temperature superconductor tapes with much higher upper critical fields are now used. Scaling is handy for extrapolating from one measurement, but may fail at low field or other temperatures, so compare with measurements over the range of use.
Use in analysis
- Collect manufacturer Jc data versus field and temperature.
- Express Jc(B, T) with scaling laws fitted to measurements.
- Compute field distribution in the coil and check margin at the worst point (peak field).
- For Nb3Sn include strain-induced Jc reduction.
- Set operating current with margin (load factor) relative to Jc at peak field.
“A coil designed at the centre field”
A prototype superconducting coil quenched at about 80% of rated current despite ample calculated margin. Design had evaluated Jc at the 7 T centre field, but the inner corner of the winding reached about 8.5 T. For NbTi that difference cuts Jc by more than 30%, and the normal transition began there. The design was revised to compute field over the whole winding and check margin at peak field.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Evaluating Jc at centre field | Quench at peak field | Evaluate at peak field |
| Scaling outside its range | Wrong Jc | Check against data |
| Comparing with copper area included | Jc understated | State non-Cu basis |
| Ignoring Nb3Sn strain | Degraded performance | Include strain dependence |
| Ignoring temperature margin | Quench from small heating | Check current-sharing temperature |
I'd like to learn related topics.
Related pages include critical current density, persistent current mode, HTS vs LTS, MRI magnet design and AC loss in superconductors.
Related Topics
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