Crystalline and preamorphized isotope multilayers are utilized to investigate the dependence of ion beam mixing in silicon (Si), germanium (Ge), and silicon germanium (SiGe) on the atomic structure of the sample, temperature, ion flux, and electrical doping by the implanted ions. The magnitude of mixing is determined by secondary ion mass spectrometry. Rutherford backscattering spectrometry in channeling geometry, Raman spectroscopy, and transmission electron microscopy provide information about the structural state after ion irradiation. Different temperature regimes with characteristic mixing properties are identified. A disparity in atomic mixing of Si and Ge becomes evident while SiGe shows an intermediate behavior. Overall, atomic mixing increases with temperature, and it is stronger in the amorphous than in the crystalline state. Ion-beam-induced mixing in Ge shows no dependence on doping by the implanted ions. In contrast, a doping effect is found in Si at higher temperature. Molecular dynamics simulations clearly show that ion beam mixing in Ge is mainly determined by the thermal spike mechanism. In the case of Si thermal spike, mixing prevails at low temperature whereas ion beam-induced enhanced self-diffusion dominates the atomic mixing at high temperature. The latter process is attributed to highly mobile Si di-interstitials formed under irradiation and during damage annealing.