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curves. We refer to this orientation as the YY configura- tion. It is observed that in the ZZ configuration, the dif- ference between the potential barriers for oblate and pro- late deformations is larger than the difference in the YY configuration. Any orientation in between these two con- figurations leads to a difference in barrier heights, which is smaller than that of the ZZ configuration and larger than that of the YY configuration. Hence, the ZZ and YY configurations represent the extreme states of bar- rier fluctuations due to surface vibrations. When the off- plane orientations are included (Ω 3 ̸ = 0), the difference in barrier fluctuations becomes minimum for the configura- tion where vibrations are perpendicular to the direction of the relative motion and perpendicular to each other. This orientation is referred to as the XY configuration, which is not included in Fig. 2. It is clear that different orientations of the nuclear surface vibrations have a very large influence on the fusion barrier fluctuations. There- fore, we need to calculate any observables by averaging over all possible relative orientations of the vibration di- rections. We carry out this averaging by sampling all three angles, { Ω 1 , Ω 2 , Ω 3 } , from a uniform distribution in the interval [0,2 π ]. 

Coupled-channels calculations are often employed for describing fusion cross-sections at sub-barrier energies. These investigations indicate that low-lying surface modes such as 2 + and 3 − make the dominant contri- bution to sub-barrier cross-sections [29–31]. Retaining only these two modes, we carry out stochastic simula- tions to describe the fusion process of Nickel isotopes. In order to compare our results with that of coupled- channels calculations of Nobre et al. [32], we adopt the same parameters as in that reference. It is important to note that none of these parameters are adjustable. The quadrupole ( λ = 2) and octupole ( λ = 3) deformation parameters are β 2 = 0 . 215, β 3 = 0 . 263 for $^{64}$Ni, and β 2 = 0 . 205, β 3 = 0 . 235 for $^{58}$Ni. The excitation ener- gies are E ⋆ 2 = 1 . 35 MeV, E ⋆ 3 = 3 . 56 MeV for $^{64}$Ni, and E ⋆ 2 = 1 . 45 MeV, E ⋆ 3 = 4 . 48 MeV for $^{58}$Ni. The zero- range potential strength and the diffuseness parameter are V 0 = − 456 MeVfm 3 and a = 0 . 56, respectively [25]. For simplicity, we consider that the quadrupole and oc- tupole vibrations of each nucleus are aligned in the same direction. 

We calculate the fusion cross section using the stan- dard expression, 

where E and P$_{l}$ ( E ) represent the incident center of mass bombarding energy and the partial transmission probabilities, respectively. Fig. 3 shows the results of the stochastic calculations and comparisons with the 

## III. FUSION CROSS-SECTIONS 

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coupled-channels calculations. The solid line corresponds to the cross section obtained by averaging over all orien- tations. The shaded area in the figure illustrates the cross section fluctuations due to the effect of different vibration orientations. The upper (lower) boundary of the shaded area corresponds to the cross section when vibrations are along (perpendicular to) the relative mo- tion direction which gives rise to maximum (minimum) cross-section due to the largest (smallest) barrier fluc- tuations. The coupling to surface vibrations increases the fusion cross-sections at sub-barrier energies, whereas it decreases the cross-sections at over-barrier energies. This effect is further enhanced due to the orientation configurations which can be easily visualized by looking at Fig. 3. At very low bombarding energies, the stochas- 

