Quasi-classical model for real space shape of the Cooper pair probabilitydensity

Within Bardeen-Cooper-Schrieffer theory of superconductivity, two electrons form the Cooper pair in the momentum space. However, it is a challenging task to represent the Cooper pair probability density in real space. Here we proposed a quasi-classical three dimensional model for the Cooper pair probability density shape in a real space, which is appeared as a direct consequence to describe the Meissner-Ochsenfeld critical field, Bc,MO (which is the thermodynamic critical field in Type-I superconductors, and lower critical field in Type-II superconductors) by the equation Bc,MO= 11/2μ0nμBln(1+√2ξ/λ), where μ0 is the magnetic permeability of free space, n is the Cooper pair bulk density, μB is the Bohr magneton, and λ is the London penetration depth, and ξ is the coherence length. As a result, the Meissner-Ochsenfeld field can be defined as the field at which each Cooper pair exhibits the diamagnetic moment of one Bohr magneton with a multiplicative pre-factor of 1/2ln(1+√2λ/ξ). Based on quasi-classical interpretation of this result, in this study we proposed that the probability density of the Cooper pair in real space can be represented as a toroid with an inner radius ξ and an outer radius of ξ + √2λ. This means that the superconducting transition is associated not only with the charge carrier pairing, but that the pairs exhibit a new topological state with genus 1.