Coherence lengths for three-dimensional superconductors in the BCS-Bose picture

Following an approach similar to that of Miyake or Randeria, Duan, and Shieh in two dimensions, we study a three-dimensional many-fermion gas at zero temperature interacting via some short-ranged two-body potential. To accommodate a possible singularity (e.g., the Coulomb repulsion) in the interaction, the potential is eliminated in favor of the two-body scattering t-matrix, the low-energy form of which is expressible in terms of the s-wave scattering length a(s). The BCS gap equation for s-wave pairing is then solved simultaneously with the number equation in order to self-consistently obtain the zero-temperature BCS gap Delta as well as the chemical potential mu, as functions of the dimensionless coupling variable lambda=k(F)a(s), where k(F) is the Fermi momentum. Results are valid for arbitrary coupling strength, and in the weak coupling limit reproduce the standard BCS results. Finally, root-mean-square pair sizes are obtained as a function of lambda and compared with experimental values.