# Surface Ripple Formation by Bombardment with Clusters: Influence of Mass
## 3 Results and Discussion
### Distribution of Volumes  
The specific weight of sputtering and redistribution and the mass effect can be studied by analyzing the volume distribution during bombardment. In the following calculations, the volume occupied by the implanted cluster atoms is neglected because their number is small compared to the rest of the atoms of the solid involved in the process. To do this, a series of quantities related to the volumes of atoms during the bombardment process are defined. If \(N_{sp}\) is the number of sputtered atoms, the volume occupied by the extracted atoms will then be \(V_{sp}=N_{sp}\cdot V_{at}\), where \(V_{at}\) is the atomic volume of a Co atom in the FCC lattice. If the number of atoms placed above the surface or adatoms is \(N_{ad}\), the volume of adatoms is then \(V_{ad}=N_{ad}\cdot V_{at}\). The sum of these two volumes would be the volume of the groove \(V_{gr}\) if the atoms of the rest of the solid did not intervene in the process. However, the atoms of the solid can fill part of the volume of the groove. In short, any atom that occupies a volume in the groove can end up being extracted, forming part of an additional layer, or displaced internally. In this way, it is verified: \(V_{gr}=V_{sp}+V_{ad}+V_{rd}\), where \(V_{gr}\) is the volume of the groove (the volume located below the surface) and \(V_{rd}\) is the redistributed volume, which turns out to be negative since the solid fills the groove. In the reference [9], another different groove volume is introduced as the sum of two volumes. \[V_{groove}=V_{no-redist}+V_{redist}=V_{sp}-V_{ad}+V_{redist}, \tag{3}\]  
where \(V_{redist}\) is the volume associated with the redistribution processes, and \(V_{no-redist}\) is the volume associated with the extraction processes. The latter volume is related to both the volume associated with the sputtering and with the adatoms. The following redistribution factor would measure the weight of redistribution in the process:  
\[f_{rd}=\frac{V_{redist}}{V_{groove}}=\frac{V_{rd}+V_{nd}}{V_{gr}-V_{ad}}. \tag{4}\]  
In Figure 6, this factor is represented for bombardment with different atoms and angles of incidence. In general, the effect of redistribution decreases with the angle of incidence. Furthermore, for any angle of incidence, the weight of redistribution is greater for heavier atoms and smaller for lighter atoms. This factor becomes practically zero for the grazing angles. In this sense, it coincides with the results obtained on metallic substrates in reference [9], which highlighted the importance of extraction processes in the surface ripple formation process. In the case of light ions, this fact also occurs for 60\({}^{\circ}\). Therefore, in these cases, the surface ripple formation process is also dominated by sputtering. The exact model establishes a value of \(f_{rd}=0.5\), below which the extraction effects would begin to predominate. However, in view of all the previous results (i.e., sputter yield and displacement), this value would be somewhat lower (around 0.2). For He, the internal redistribution of the atoms would lead to the formation of the added layers so that both quantities, \(V_{ad}\) and \(V_{rd}\) (<0), would approximately compensate (\(V_{redis}\sim 0\)). However, the same does not occur with the difference in volumes of extracted material (\(V_{sp}\)) and additional layers (\(V_{ad}\)), which would be negative (\(V_{no-redist}<0\)). In any case, it is verified that \(V_{groove}<0\) and the main contribution to this volume is due to the non-redistributed volume of negative origin, that is, induced by the formation of the added layers. Therefore, it should not be considered as an extraction effect. In summary, this graph confirms that the formation of nanopatterns on metallic substrates is explained as a function of the angle of incidence. Thus, for intermediate angles, the underlying effect is atomic redistribution, and for grazing angles, it is sputtering. For any angle of incidence, the weight of redistribution increases with mass.