Cylindrically Symmetric Solutions of the Ginzburg-Landau Equations

The Ginzburg-Landau questions have been solved for an isolated magnetic flux line enclosing a single flux quantum. The radial dependence of the magnetic field $H$, the order parameter $n_e$, the current density $J$, and the resulting free energy per unit length $\frac{F}{L}$, are obtained for values of the Ginzberg-Landau parameter $\kappa =20, 5, 1.0, 0.5, \mathrm{and} 0.2$. For $\kappa \le 5$, the axial magnetic field $H\left(0\right)\mathrm{}$ is approximately given by $H\left(0\right)=0.6\mathrm{}{\kappa }^{-0.43\mathrm{}}\sqrt{2}{H}_{\mathrm{cb}}$. The maximum value of the current density is approximately $J=\frac{0.33\sqrt{2}{H}_{\mathrm{cb}}}{\lambda }$, where $H_{\mathrm{cb}}$ is the bulk critical field and $\lambda$ is the superconducting penetration depth.