Summation curves for flux pinning in superconductors

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Abstract
Flux pinning in irradiated superconductors having either dislocation-loop or void microstrutures is reexamined. The elementary interaction force fp can be accurately computed for such defects if their size is known from transmission electron microscopy. Data exist for a range of defects for which fp varies by over a factor of 104. If the data for voids and dislocation loops are plotted as Q, the volume pinning force per defect, versus fp at a given reduced induction b, the sets of data for each defect species overlap to form a single summation curve. This curve is not correctly described by either of the two most widely used summation models, the direct summation or the statistical summation of Labusch. The empirical summation curve runs parallel to, but a factor ?4 below, the direct summation curve at high fp but bends to decrease approximately as f2p at low fp. Increasing b produces a shift in the summation curve, parallel to the direct summation Q=fp line, by a factor which is proportional to (1−b)2. This shift is the origin of the peak effect which appears at high b for low fp defect microstructures. The failure of the statistical theory is thought to result from the presence of flux-line dislocations in the flux-line lattice which produce both a large decrease and a dispersion in the local threshold force necessary for pinning centers to be effective.