Characteristic length for pinning force density in Nb3SnNb{_3}Sn

The pinning force density $F{_p}(J{_c},B)=J{_c} \times B$ (where $J_c$ is the critical current density, $B$ is applied magnetic field) is one of main quantities which characterizes the resilience of a superconductor to carry dissipative-free transport current in applied magnetic field. Kramer (1973 J. Appl. Phys. 44 1360) and Dew-Hughes (1974 Phil. Mag. 30 293) proposed a widely used 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 late 1970-s till now, several research groups reported experimental data for the dependence of $F_{p,max}$ on the average grain size, $d$, in $Nb{_3}Sn$-based conductors. Godeke (2006 Supercond. Sci. Techn. 19 R68) proposed that the dependence obeys the law $|F{_{p,max}}(d)|=A \times log(1/d)+B$. However, this scaling law has several problems, for instance, the logarithm is taken from a non-dimensionless variable, and $|F{_{p,max}}(d)|< 0$ for large grain sizes and $|F{_{p,max}}(d)|\rightarrow \infty$ for $d \rightarrow 0$. Here we reanalysed full inventory of publicly available $|F{_{p,max}}(d)|$ data for $Nb{_3}Sn$ conductors and found that the dependence can be described by $F_{p,max}(d)= F_{p,max}(0)exp(-d/{\delta})$ law, where the characteristic length, ${\delta}$, is varying within a remarkably narrow range, i.e. ${\delta}=(175 \pm 13) nm$, for samples fabricated by different technologies. The interpretation of the result is based on an idea that the in-field supercurrent is flowing within a thin surface layer (the thickness of ${\delta}$) near the grain boundary surfaces. Alternative interpretation is that ${\delta}$ represents characteristic length for the exponentially decay flux pinning potential from dominant defects in $Nb{_3}Sn$ superconductors, which are grain boundaries.Comment: 22 pages, 8 figure