Hysteresis in Superconducting Alloys—Temperature and Field Dependence of Dislocation Pinning in Niobium Alloys

Critical current and pinning-force densities in a series of niobium alloys subjected to severe plastic deformation have been determined from measurements of complete hysteretic magnetization curves on alloys with Ginzburg-Landau parameter $\kappa$ between 1.3 and 13 at temperatures from the critical temperature $T_c$ down to $0.14T_c$. Systematic scaling rules were found that accurately describe all of the results over the entire range of fields and temperatures. The pinning-force density scales with magnetic induction as a single function of $\frac{B}{H_{c2\mathrm{}}}$; it scales with temperature as the $\frac{5}{2}$ power of the upper critical field $H_{c2\mathrm{}}\mathrm{}\left(T\right)\mathrm{}$, is roughly proportional to ${\kappa }^{-\gamma }$, where $1<\mathrm{}\gamma <3\mathrm{}$, and is otherwise independent of $T$. A model for the pinning process that takes into account deformation of the fluxoid lattice by the pinning forces is proposed to account for the observed scaling rules. The results are consistent with a pinning interaction based on a second-order elastic interaction between dislocations and the fluxoid lattice, but other mechanisms are not excluded. Cooperative effects seem to be an essential feature of the pinning process, leading to a dependence of the pinning-force density on the square of the pinning-point strength, and on the arrangement of pinning points.