The fusion neutron evaporation cross sections induced by (superscript 4,6,8)He on a 209Bi target were studied at energies near the Coulomb barrier. Our investigations concentrated on the effect of radioactive neutron exotic nuclei like (superscript 6,8)He on the neutron evaporation cross sections, comparing with their stable isotopes like 4He. The calculations of the theoretical cross sec- tions were done using the Code PACE4 with r0=1.3 fm, and V0=67 MeV for all the different types of reactions. The transmission probability for a one-dimensional barrier depended on quantum mechanical theory. Our calculations were compared with different experimental results obtained from JINR, Russia. Good agreement was obtained between the experimental results and the theoretical calculations. The enhancement in the neutron evaporation cross sections increased with increasing the number of weakly bound neutrons of radioactive nuclei, such as 8He and 6He, compared with the stable nuclei 4He.