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The pinning force density $F{_p}(J{_c},B)=J{_c} \times B$ (where $J_c$ is thecritical current density and $B$ is the magnetic field) is one of the mainparameters that characterize the resilience of a superconductor to carry adissipative-free transport current in an applied magnetic field. Kramer (1973J.Appl.Phys. 44 1360), and Dew-Hughes (1974 Phil.Mag. 30 293) proposed a widelyused scaling law for the pinning force density amplitude:$F{_p}(B)=F{_{p,max}}((p+q){^{(p+q)}}/({p^p}{q^q}))(B/B_{c2}){^p}(1-B/B{_{c2}})^q$,where $F{_{p,max}}$, $B{_{c2}}$, $p$, and $q$ are free-fitting parameters.Since the late 1970-s till now, several research groups have reportedexperimental data on the dependence of $F_{p,max}$ on the average grain size,$d$, in $Nb{_3}Sn$-based conductors. Godeke (2006 Superc.Sci.Techn. 19 R68)proposed that the dependence obeys the law $|F{_{p,max}}(d)|=A \times ln(1/d)+B$. However, this scaling law has several problems, for instance, the logarithmis taken from a non-dimensionless variable, and $|F{_{p,max}}(d)|< 0 $ forlarge grain sizes and $|F{_{p,max}}(d)|\rightarrow \infty $ for $d \rightarrow0$. Here we reanalysed the full inventory of publicly available$|F{_{p,max}}(d)|$ data for $Nb{_3}Sn$ conductors and found that the dependencecan be described by $F_{p,max}(d)= F_{p,max}(0)exp(-d/{\delta})$ law, where thecharacteristic length, ${\delta}$, varies within a remarkably narrow range,that is, ${\delta}=(175 \pm 13) nm$, for samples fabricated by differenttechnologies. The interpretation of the result is based on the idea that thein-field supercurrent flows within a thin surface layer (thickness of${\delta}$) near the grain boundary surfaces. An alternative interpretation isthat ${\delta}$ represents characteristic length of the exponential decay fluxpinning potential from the dominant defects in $Nb{_3}Sn$ superconductors,which are grain boundaries.

Characteristic length for pinning force density in $Nb{_3}Sn$