Simple model for the vortex core in a type II superconductor

In order to model the core of an isolated vortex in a type II superconductor, a variational trial function for the magnitude of the normalized order parameter of the form f = ϱ/R is assumed, where ϱ is the radial coordinate, R = (p ² + ξ_v ²)^1/2, and ξ _v is a variational core radius parameter. Remarkably simple analytic expressions for the magnetic flux density and supercurrent density that solve Ampere's law and the second Ginzburg-Landau equation are obtained. An analytic result for the free energy of the isolated vortex is then derived by integrating the Ginzburg-Landau free energy functional. The value of ξ _v that minimizes the free energy is calculated as a function of the Ginzburg-Landau parameter κ = λ/ξ and is found to range from ξ _v = 0.935ξ for κ = 0.707 to ξ _v ≈ 1.414ξ for κ > 1. A simple expression for the form factor or the Fourier transform of the flux density is obtained, which may be useful in the analysis of neutron diffraction experiments.