Numerical Constants for Isolated Vortices in Superconductors

Abstract

For isolated vortex lines in high-$\ensuremath{\kappa}$, type-II superconductors, Abrikosov derived the expressions $B(0)={\ensuremath{\kappa}}^{\ensuremath{-}2}(\mathrm{ln}\ensuremath{\kappa}+{C}_{0}){H}_{c2}$ and ${H}_{c1}=\frac{1}{2}{\ensuremath{\kappa}}^{\ensuremath{-}2}(\mathrm{ln}\ensuremath{\kappa}+{C}_{1}){H}_{c2}$, but the numerical values he provided for the constants ${C}_{0}$ and ${C}_{1}$ were previously found to violate an identity ${C}_{1}\ensuremath{-}{C}_{0}\ensuremath{-}\frac{1}{2}={C}_{\ensuremath{\gamma}}>0$. The constants are reevaluated, giving ${C}_{0}=\ensuremath{-}0.282$, ${C}_{1}=0.497$, ${C}_{\ensuremath{\gamma}}=0.279$. Furthermore, for superconductors containing a high concentration of magnetic impurities, it was previously shown that the electric field generated at the center of an isolated vortex in flux-flow situations is proportional to ${H}_{c2}$ and the flux-flow velocity $v$. The proportionality constant ${C}_{E}$ in the high-$\ensuremath{\kappa}$ limit is numerically evaluated here to be 0.951, which, together with the value for ${C}_{\ensuremath{\gamma}}$, determines the flux-flow resistivity ${\ensuremath{\rho}}_{f}=0.381{\ensuremath{\rho}}_{n}\frac{〈B〉}{{H}_{c2}}$ in the low-applied-field limit when vortices are very far apart.