7 of non-fusion inelastic collisions have been estimated by averaging over the ensemble of trajectories. The simple stochastic semi-classical approach provides a surprisingly good agreement with quantal coupled-channels calcula- tions as well as the experimental data for fusion cross- sections of nickel isotopes at sub-barrier and near-barrier energies. Of course, barrier penetration is not included in the description. However, at near-barrier energies, the dominant effects on fusion arise from the barrier fluctu- ations, which are well accounted for by the stochastic approach. nally by Esbensen et al. [14], we carry out simula- tions for describing fusion and some other properties of the collision of nickel isotopes at sub-barrier and near- barrier energies. As known from coupled-channels calcu- lations, dominant effects for describing sub-barrier fusion arise from the coupling of the relative motion with the low-lying surface vibrations. Therefore, in analogy to coupled-channels calculations, we consider the coupling of the relative motion with low-lying collective surface vi- brations. Since the de Broglie wavelength is very short, a classical treatment provides a good approximation for the relative motion. However, the qauntal aspects of sur- face vibrations play a dominant role in the sub-barrier fusion mechanism. In the present approach, we incor- porate quantal zero-point fluctuations of the surface vi- brations in a stochastic approximation. In the applica- tions presented here, only the quadrupole and octupole vibrations of the projectile and target nickel ions are in- cluded. An ensemble of trajectories of the relative motion are generated by picking the initial conditions of surface vibrations according to the quantal zero-point fluctua- tions of their ground states. Some observables such as fusion cross-sections and final kinetic energy distributions S.A. and B.Y. gratefully acknowledge GANIL, and also S.A. and D.L. gratefully acknowledge TUBITAK and Middle East Technical University for their support and warm hospitality extended them during their visits. This work is supported in part by the US DOE Grant No. DE- FG05-89ER40530. Acknowledgments FIG. 8: The distributions of the fusion time in collisions of $^{64}$Ni nuclei are plotted for different incident center of mass energies (a) and different angular momenta (b). For each plot the distribution function is normalized with the total number of fusion events. FIG. 7: The average values of the total excitation energy (a) and the final energy (b) in head-on collisions of $^{64}$Ni nuclei are plotted as a function of the incident center of mass energy. The excitation energy is given by E ⋆ = E inc − E final . 2 4 6 8 10 12 Excitation Energy (MeV) (a) 90 91 92 93  90  92  94  96  98 100 102 104 Final Energy (MeV) Incident Energy (MeV) (b) 0.1 0.2 0.3 0.4 0.5 of fusion events l = 0 (a) E$_{inc}$ 90 MeV E$_{inc}$ 92 MeV E$_{inc}$ 94 MeV E$_{inc}$ 96 MeV E$_{inc}$ 98 MeV E$_{inc}$ 100 MeV 0.0 0.1 0.2  0.8  1.2  1.6  2  2.4 normalized number fusion time (10$^{-21}$ s) E$_{inc}$ 96 MeV (b) l = 40 h l = 30 h l = 0