# Surface Ripple Formation by Bombardment with Clusters: Influence of Mass
## 3 Results and Discussion
### _Sputter Yield_  
As mentioned above, there are two collisional processes that induce the formation of surface waves: sputtering and redistribution. Sputtering allows the surface to be undermined, further highlights its morphology, and deepens the groove. Next, the sputter yield \(Y\) and the surface distortion are related. Figure 4 represents the angular dependence of sputtering for bombardment with clusters of different types of atoms. The proportionality of the sputter yield with the function \((cos\theta)^{-5/3}\) in the bombardment with atoms where \(\theta\) is the angle of incidence is verified only for an incidence that is not too oblique and \(M_{2}/M_{1}<3\), where \(M_{2}\) is the mass of Co and \(M_{1}\) is the mass of the projectile atom [24]. For He, the proportionality is with the function \((cos\theta)^{-1}\). Furthermore, there is another proportionality factor that depends on the stopping power. This parameter, in turn, is an exponential function of the mass ratio \(M_{2}/M_{1}\). That is, the greater the mass of the projectile atom, the lower stopping power and, therefore, the lower sputter yield. This trend is also observed in Figure 4 for angles below the maximum for cluster bombardment. The sputtering curves have a more or less pronounced maximum, depending on the mass of the cluster atom. It is very abrupt in the curves of Xe and Kr and very smooth in those of Ne and Ar. In fact, this maximum depends essentially on the surface structure, as indicated by experiments [25]. Likewise, it is also difficult to make an approximate analytical prediction of the values at grazing incidence.  
Figure 4: Sputter yields for different types of clusters (He, Ne, Ar, Kr, and Xe) as a function of the angle of incidence. The accounting is performed by incident atom. The average value is taken after bombarding with 167 clusters.  
Figure 3: Comparison between the local atomic temperatures after 1 ps from the impact of cluster number 168 for (**a**) Ne and (**b**) Xe. The coldest atoms have blue hues, and the hottest atoms have red hues. Only a 10 Å thick fringe parallel to the \(xz\) plane around the impact point is shown. The angle of incidence is 50\({}^{\circ}\). The temperature is expressed in K.  
In the case of He, inelastic collisions cannot be neglected for any value of the initial bombardment energy, so the previous results would not apply to this case [26]. It is not even possible to use an amorphous solid model, such as the one that allows the deduction of the previous results [24], since channeling effects are important for non-grazing incidences. For example, at an angle of 20\({}^{\circ}\), 29% of the He atoms pass through the solid. Both of these factors radically reduce sputtering; however, the latter is especially important.  
Regarding the surface morphology formation, atom extraction is greater for heavier cluster atoms in the case of grazing incidence (Figure 4). These types of atoms are the ones that form the most prominent surface patterns. Therefore, it is clear that sputtering would play a more decisive role only for grazing angles (\(\sim\)70\({}^{\circ}\)) and even for close angles (\(\sim\)60\({}^{\circ}\)), but not in all the other cases. In the case of heavy atoms, the cascade energy is deposited deeper for non-grazing incidence, making it difficult for atoms to leave the solid and, therefore, not contributing to the formation of patterns.