# Pattern Formation on Alloys by Cluster Ion Beam Irradiation
## 3 Results and Discussion
### Relocation Cross-Sections  
We analyze the phenomenon of ripple formation from the point of view of the relocation cross-sections [34]. In our case, we have calculated the relocation cross-section (outward), defined as the probability that an atom initially located at position (\(x\), \(dx\)) is relocated to position (\(x+z\), \(dz\)) after a fluence \(\delta\phi\). From this section, the probability can be directly obtained that an atom initially at position (\(x-z\), \(dz\)) is relocated to (\(x\), \(dx\)) after fluence \(\delta\phi\) (inward section). In summary, sections are obtained by counting the atoms that change position and dividing them by the layer width, density, and fluence. Figure 7 shows the inward and outward sections for the pure Cu substrate and the alloy of equal concentration. The cross-sections are symmetric at short distances for the crystalline material (Figure 7a). In the case of amorphous material, short-range material filling imbalances occur (Figure 7b). In contrast, while the crystalline material facilitates structure formation via outward displacement from the output layer, the amorphous material leads to short-range void and interstitial formation by atoms entering and leaving the layer. In short, the greater number of defects generated in amorphous solids increases the local mobility of atoms. However, this mobility is short-range and does not affect the process in question.  
Figure 6: Number of displaced particles of each species for a 50% Au-Cu alloy bombarded at 70\({}^{\circ}\). The figure illustrates the distribution of atoms in ridges, sputtered atoms, and those redistributed within the material.  
From the cross-sections, all the atoms that arrive or leave a given layer can be calculated. Figure 8 shows sputtered atoms and outward and inward relocated atoms for the Cu substrate and for the alloy of equal concentration bombarded at 50\({}^{\circ}\) and 70\({}^{\circ}\). In this case, we will be interested in analyzing it in vertical layers. In the 70\({}^{\circ}\) curves, the greatest weight of sputtering is observed in the case of Cu (Figure 8b). In fact, in both cases, sputtering is the dominant factor. In the curves, for example, of 50% Au-Cu (Figure 8a), it can be observed how the difference between the relocated atoms (those that leave minus those that arrive) may even turn out to be locally greater than that of sputtered atoms. However, the net balance of relocated atoms along the entire \(z\)-axis does not exceed the total number of sputtered atoms. This fact occurs in both cases for the alloy (70\({}^{\circ}\), Figure 8a, and 50\({}^{\circ}\), Figure 8c) and in the case of pure Cu bombarded with a grazing angle, that is, in the cases in which sputtering predominates (Figure 8b). We must understand that the difference in the number of atoms that arrive and leave a layer is a measure of the redistribution effect. Furthermore, this effect is especially important in the case of pure Cu bombarded at intermediate angles (Figure 8d), where the total number of sputtered atoms does not exceed the total number of redistributed atoms [20]. Redistribution also induces differences in amorphous materials. Thus, in Figure 8c,d, it is observed that, although the bottom of the groove is emptied, this same bottom is filled again with atoms from other layers. However, this effect does not occur in crystalline materials, where that material rises vertically to form the ridges.  
Figure 7: Horizontal relocation cross-sections for position \(x=0\) Å and an angle of incidence of 50\({}^{\circ}\) and two surfaces of different composition (at.% Cu): (**a**) 100 and (**b**) 50 as a function of the final position (outward section) or the initial position (inward section).