Calculation of the elastic scattering parameters in an ultra-cold   Fermi-Bose and Bose-Bose Fr vapor
  The calculations of the elastic scattering properties of ultracold-francium isotopes are reported in the detail. A parametric model potential for the triplet molecular states of the Fr$_{2}$ is represented, and the scattering lengths at and the effective range $r_{e}$ are calculated using WKB and Numerov methods for Fr-Fr in the triplet state. The convergence of these scattering properties as the depending on a $K_{0}$ parameter and core radius is also investigated using two method as Quantum Defect Theory and analytical formula of symskovisky. 
M. Kemal \"Ozt\"urk, S\"uleyman \"Oz\c{c}elik