Numerical Constants for Isolated Vortices in Superconductors

For isolated vortex lines in high-$\kappa$, type-II superconductors, Abrikosov derived the expressions $B\left(0\right)={\kappa }^{-2\mathrm{}}(\ln \kappa +C_0)H_{c2\mathrm{}}$ and $H_{c1\mathrm{}}=\frac{1}{2}{\kappa }^{-2\mathrm{}}(\ln \kappa +C_1)H_{c2\mathrm{}}$, but the numerical values he provided for the constants $C_0$ and $C_1$ were previously found to violate an identity $C_1-C_0-\frac{1}{2}=C_{\gamma }>0\mathrm{}$. The constants are reevaluated, giving $C_0=-0.282\mathrm{}$, $C_1=0.497\mathrm{}$, $C_{\gamma }=0.279\mathrm{}$. 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 $H_{c2\mathrm{}}$ and the flux-flow velocity $v$. The proportionality constant $C_E$ in the high-$\kappa$ limit is numerically evaluated here to be 0.951, which, together with the value for $C_{\gamma }$, determines the flux-flow resistivity ${\rho }_f=0.381\mathrm{}{\rho }_n\frac{〈B〉}{H_{c2\mathrm{}}}$ in the low-applied-field limit when vortices are very far apart.