Scaling laws for flux pinning in hard superconductors

For all hard high-field superconductors examined to date, there is a maximum in the pinning force density Fp as a function of the reduced magnetic field h. Fietz and Webb first demonstrated in dilute Nb alloys that the peak in Fp scales as [Hc2(T)]2.5 if the temperature is changed; the maximum value of Fp occurred at the same value of reduced field regardless of temperature. Recent data on the temperature dependence of pinning in Nb3Sn, Nb–25% Zr and a Nb–Ti alloy, which exhibits the ``peak effect'', are analyzed to show that similar scaling laws are obeyed by these materials. All presently available evidence indicates however that the reduced field hp at which the maximum Fp occurs, as well as the height and shape of this maximum, can be altered by metallurgical treatment. Apparently weak pinning defects, or widely spaced ones, produce a small peak in Fp(h) at high h whereas strong closely spaced pins produce a large peak in Fp(h) at low h without producing much change in Fp(h) at high h. A model which predicts these metallurgical effects, as well as the scaling laws, is proposed. According to the model at h≪hp, flux motion occurs primarily by unpinning of line pins, whereas at h≫hp it occurs by synchronous shear of the flux line lattice around line pins too strong to be broken. In the high-field regime, where quantitative predictions are possible, the magnitude and field dependence of Fp are in good agreement with experiment. In this model the anomalous ``peak effect'' occurs whenever line pins are relatively weak, producing a narrow peak in Fp(h) at high h.