BCS-Bose model of exotic superconductors: Generalized coherence length

Analytic expressions are derived for the root-mean-square (rms) radius of a pair of fermions in a BCS many-fermion state in one, two, and three dimensions, in terms of the BCS gap energy and the associated chemical potential. These expressions are valid for any coupling strength of any pair interaction model implying a momentum-independent gap energy. The latter holds, e.g., for an attractive δ pair potential examined in the one-dimensional (1D) case (whose N-fermion ground state can be determined exactly) or for the BCS (electron-phonon) model interaction in any dimension. Weak-coupling and/or high-density limits for the rms radius are identical in 1D, 2D, and 3D, and reduce to the familiar well-known Pippard result to within a factor of order unity. In contrast, strong-coupling and/or low-density limits coincide in 1D and 3D, but differ by a factor of order unity in the 2D limit, and in each case are essentially the size of a single, isolated pair. The 1D δ interaction McGuire-Yang-Gaudin many-fermion model is studied in detail. The interaction renormalization scheme of Miyake and of Randeria, Duan, and Shieh, and the BCS interaction model, both in 2D, are employed to analyze cuprate superconductor empirical results. Reasonable agreement between theoretical rms radii with experimental coherence lengths suggests that cuprates can be described moderately well as weakly coupled superconductors within the BCS-Bose formalism