Studying interactions of radioactive ions with neutrons is particularly demanding and has been performed only in a few cases. Some of these interactions are crucial in several astrophysical contexts. In the present work, the case of the 7 Be destruction induced by the (n, α) reaction is investigated at the energies typical of the primordial nucleosynthesis by means of the Trojan Horse Method applied to the 2 H (7 Be, αα) p quasi-free reaction. The 7 Be (n, α) 4 He cross-section has been measured in a single experiment from∼ 2 MeV down to cosmological energies. The corresponding deduced reaction rate has been adopted to evaluate the impact on big bang nucleosynthesis and on the lithium problem.