³S^e and ¹S^e scattering lengths for e^- + Rb, Cs and Fr collisions

We calculate the ³S^e and ¹S^e scattering lengths in low-energy e^- + Rb(5s), Cs(6s), and Fr(7s) collisions. Our approach is based on the relativistic version of the modified effective range theory (MERT) and extrapolates eigenphases provided by Dirac R-matrix calculations to zero energy. To test the accuracy of our MERT extrapolation method, we compare the fine-structure components for the lowest ³P^o resonance of Fr^- ion with full Dirac R-matrix calculations.