NUMERICAL-CALCULATION OF FLUX PINNING BY ALPHA-TI PRECIPITATES IN NB-TI

We have numerically calculated the elementary pinning force, f(p) and the bulk pinning force, F(p), for Nb-Ti alloys containing alpha-Ti ribbon-like precipitates. The Nb-Ti microstructure is modelled as an array of parallel slabs of normal alpha-Ti of thickness d(n) < 5-xi separated by superconducting Nb-Ti of thickness d(s) < 5-xi. We solve the Ginzburg-Landau (G-L) equations, for each alloy, to obtain the order parameter PHI(o)(r) within the structure. The fluxon-pin interaction is calculated by overlapping PHI(o)(r) with PHI(f)(r), the order parameter around the fluxon core, and re-evaluating the G-L free energy. Repeating this process as a function of fluxon position, we can map out the pinning potential and calculate both f(p) and F(p) for each model alloy. f(p) is found to increase steadily but non-linearly for d(n) < 1.5-xi and plateau for d(n) > 3-xi. Additionally, we note an enhancement in f(p) for 1.5-xi < d(n) < 3-xi due to the proximity effect. Our F(p) calculations predict a maximum for an alloy with d(n) = 1.0-xi and d(s) = 1.6-xi. This result suggests that a Nb-Ti alloy with 40vol% alpha-Ti and 6 nm thick ribbons would maximize F(p) and yield an increase in J(c) of 40% over present alloys. In addition, our theoretical results for F(p) closely parallel the experimental results of Meingast and Larbalestier[1] for F(p) as a function of the pin thickness and the results of Lee and Larbalestier for F(p) as a function of alpha-Ti volume fraction[2].