We predict dramatic resonances in low-energy electron elastic total cross-sections for Rb, Cs and Fr atoms, whose energy positions are identified with the electron affinities (EA's) for these atoms, preceded by shape resonances and Ramsauer-Townsend minima. This provides a new powerful theoretical method of determining unambiguous EA values for atoms. The extracted EA values for Rb and Cs atoms agree very well with the most recently measured values, but our value for Fr disagrees significantly with existing calculated values. The calculation used the recent Regge-pole methodology with a Thomas-Fermi potential incorporating the vital core-polarization interaction. (C) 2008 Published by Elsevier B.V.