# Pattern Formation on Alloys by Cluster Ion Beam Irradiation
## 3 Results and Discussion
### Volume Exchange  
In the following, we analyze the process of volume exchange during atomic bombardment. In principle, we disregard the volume of the Ar atoms we implant as being small compared to the rest. In general, any atom that occupies a volume in the groove can end up sputtered, redistributed, or placed on the surface. Therefore, we can affirm that: \(V_{gr}=V_{sp}+V_{ad}+V_{rd}\), where \(V_{ad}\) is the volume occupied by the atoms in the ridges, \(V_{sp}\) is the volume of the eroded atoms, and \(V_{rd}\) is the redistributed volume of negative value because the atoms of the solid generally fill the groove. The redistribution of atoms is associated with the formation of ridges and the internal rearrangement of atoms, that is, \(V_{\text{redist}}=V_{ad}+V_{rd}\). This causes the groove volume to be created from redistributed and unredistributed atoms [28]. \[V_{gr}=V_{\text{no-redist}}+V_{\text{redist}}=V_{sp}+V_{ad}+V_{rd} \tag{2}\]  
We could define a redistribution factor as shown in the following expression:  
\[F_{rd}=\frac{V_{\text{redist}}}{V_{gr}}=\frac{V_{ad}+V_{rd}}{V_{gr}} \tag{3}\]  
At this point, we must remember that a redistribution factor greater than 0.2 indicates a higher redistribution weight [21]. In Figure 4, we represent this factor for surfaces of different concentrations bombarded at two angles of incidence as a function of time. Figure 4a corresponding to intermediate angles (50\({}^{\circ}\)) is more interesting. This figure shows a prominent redistribution factor for pure substances. According to the value indicated previously, the effect of redistribution predominates in these materials. However, for alloys of similar concentration, this factor becomes small and even negative, which means that sputtering prevails. A negative volume means that the volume located above the surface at the ridges becomes smaller than the volume that recedes to fill the groove. This fact confirms the maximum found in the sputtering yield and the minimum in the displacements. It should also be noted that the redistribution factor of light-species metals is much greater than that of heavy-species metals. This links the effect to the greater mobility of Cu. Figure 4: Redistribution factor \(F_{rd}\) for different angles of incidence: (**a**) 50\({}^{\circ}\) and (**b**) 70\({}^{\circ}\) and surfaces of varying compositions (at.% Cu): 0, 20, 35, 50, 65, 80, and 100 as a function of time. For their part, the curves for grazing angles are all very similar (Figure 4b), except for the equal concentration curve, which shows significantly negative values. According to this model, sputtering is clearly the predominant factor in all cases. This result corrects previous findings that attributed the predominance of redistribution effects to amorphous solids [28]; those findings are likely specific to the material studied, amorphous silicon. We should study the angular differences in the redistribution factor with the concentration of alloys. To do this, in Figure 5, we compare this factor in the bombardment of two surfaces: one of pure Au and the other of Au-Cu at the same concentration. In the latter case (Figure 5a), the curves indicate that sputtering is the dominant effect at all studied angles, as the factor remains below 0.2. In fact, the volumes of the ridges are very similar for all angles, with the volume of the groove differing somewhat more. In any case, the redistributed volume gives negative volumes because of the small size of the ridges. These dimensions of the ridges are probably induced by the limited vertical travel of the atoms due to the lack of crystallinity of the solid. This is evident when comparing the average displacements in the \(z\)-direction: 2.5 A for Au and \(-\)1 A for 50% Au-Cu, both bombarded at 50\({}^{\circ}\); 0.24 A for Au and \(-\)0.15 A for 50% Au-Cu, both bombarded at 70\({}^{\circ}\).