DIRECT SUMMATION OF ELEMENTARY PINNING FORCES FOR NB-TI SUPERCONDUCTORS WITH A FIELD-DEPENDENT PROBABILITY FUNCTION

As T --> T(c) the field dependence of F(p) for optimized, high-J(c) Nb-Ti changes progressively from b(1 - b) to approximately b(1 - b)2.5, a result which has no explanation. In this paper, we investigate the influence of a pinning probability function P(b) on the field dependence of F(p). The elementary pinning force (f(p)) for a core-pinning interaction may be greater than the value which would be obtained for a random sampling of the pins (<f(p)>) because the pins which have an above-average strength will be occupied with higher probability. As the field increases, the field-independent line energy becomes stronger than the decreasing pinning energy, and the deviations necessary for the selection of above-average pins occur less often. The difference between f(p) and <f(p)>, which is proportional to P(b), accordingly decreases. From direct summation, F(p) almost-equal-to (B/phi(0)). f(p), and because the field dependence of f(p) is proportional to 1 - b (due to the core interaction) and also to P(b), the field dependence of F(p) is proportional to b(1 - b)P(b). This may resolve the F(p) results.