Effect of breakup on the fusion of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mmultiscripts><mml:mi mathvariant="normal">Li</mml:mi><mml:mprescripts/><mml:none/><mml:mn>6</mml:mn></mml:mmultiscripts></mml:mrow></mml:math>,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mmultiscripts><mml:mi mathvariant="normal">Li</mml:mi><mml:mprescripts/><mml:none/><mml:mn>7</mml:mn></mml:mmultiscripts></mml:mrow></mml:math>, and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mmultiscripts><mml:mi mathvariant="normal">Be</mml:mi><mml:mprescripts/><mml:none/><mml:mn>9</mml:mn></mml:mmultiscripts></mml:mrow></mml:math>with heavy nuclei

Comprehensive high precision complete and incomplete fusion cross sections have been measured for the $^{6}\mathrm{Li}+^{209}\mathrm{Bi}$, $^{7}\mathrm{Li}+^{209}\mathrm{Bi}$, and $^{9}\mathrm{Be}+^{208}\mathrm{Pb}$ reactions, at energies near and below the Coulomb barrier. The experimental details and the analyses procedures for the characteristic $\ensuremath{\alpha}$-decay and fission measurements are described. Three different methods are used to conclusively show the large suppression of complete fusion at energies around and above the average barrier, which is associated with the projectile nuclei having a low energy threshold against breakup. First, theoretical predictions of fusion cross sections above the average barrier are compared with the data, and second the area under the measured barrier distribution is compared with expectations. The sensitivity of the suppression factors to variables which can affect the calculated cross sections is thoroughly investigated. The third method, essentially model independent, compares the complete fusion cross sections for the $^{7}\mathrm{Li}+^{209}\mathrm{Bi}$ and $^{9}\mathrm{Be}+^{208}\mathrm{Pb}$ reactions with those for the fusion of nuclei with a high threshold against breakup, which produce the same compound nucleus. All methods give consistent results, showing that the complete fusion cross sections at energies around and above the barrier are suppressed by $\ensuremath{\sim}30%$ compared with reactions of nuclei having a high energy threshold against breakup. The cross sections for incomplete fusion are found to be similar to the missing complete fusion cross sections. The experimental controversies regarding the effect of breakup on fusion is discussed, and the importance of unambiguously separating complete fusion from incomplete fusion is emphasized. This distinction is also important to achieve theoretically for realistic modelling of fusion of nuclei which break up readily.