The K-shell ionisation of atoms by fast alpha particles is investigated using the Moller relativistic modification of the Born approximation. The incident and scattered alpha particles were described by relativistic plane waves. The Darwin approximation was used for the description of the K-shell atomic electrons and the ejected electron. The total cross sections were calculated as functions of alpha -particle impact energy for Sn, Ag, Au, Pb and compared with the available experimental data and calculations. The results obtained are in good accordance with the experimental data of Dost et al (1981). The influence of spin change of the K-shell electron on the total cross section was also studied. The results show that the collisions involving spin change of the atomic electron could be significant at higher energies of the incident alpha particles, especially for atoms with greater nuclear charge Z.