## 4 Discussion and Conclusions  
Our computational results employing DFT-LDA are consistent and in sufficient agreement with previously calculated and experimental values. Especially, our results on vacancies in aluminum agree well with other ab initio calculations [15] and experiments [16] giving credence to our approach. Moreover, the inward relaxation of nearest-neighbor Al-atoms surrounding a single Cu atom is in accordance with experimental results from X-ray absorption [47]. Furthermore, we find a binding between Cu atoms situated on the \([100]\)-habit planes in fcc aluminum, while, e.g., a 3D tetrahedron even shows a repulsive interaction. Thus, 2D structures are favored compared to 3D ones, which is in agreement with experimental observations finding mono-atomic copper platelet on the \([100]\)-planes in the fcc Al-lattice when segregation of the super saturated solution is starting [41]. The calculated total energies for a 2D copper cluster vary slightly with used supercell sizes and shapes. However, the dependence of the total energy on the number of Cu atoms shows a similar trend in all cases. Nevertheless, it turns out that there have to be at least three layers of Al atoms in the supercell perpendicular to the Cu-habit plane separating periodic images of the copper platelets. Obviously, the relaxation of aluminum planes parallel to the copper platelet, which is important for  
Figure 7: Binding energies of Zn atoms in 2D and 3D Zn clusters. (**left**) The binding energy for the last attached Zn atom. (**right**) The binding energy per Zn atom in the cluster. energy lowering, cannot be reliably described with wide but flat supercells. This type of cells is not large enough in the z-direction perpendicular to the copper platelet. In contrast to magic number arrangement of defects in semiconductors [48], for metallic materials it seems to be difficult to state that distinguished structures are existing at all. While in general the binding energies for different geometric configurations differ very little (about 50-100 meV), we also find some special platelet structures of Cu atoms with quite high binding energies between the copper species. However, the generally small differences in binding energies may be responsible for not finding any special configurations in experiments. Remarkable is only that some closed configurations of Cu atoms on the [100]-planes show higher binding energies per Cu atom (four Cu arranged in a square). Obviously, their geometries allow electronic structures, which are energetically more favorable. However, it seems to be experimentally demanding to confirm this finding. Concerning 3D instead of 2D plate-like structures, such as the 3D-tetrahedron, we find that these close-packed structures are the least favorable ones at all, since the Cu atoms inside their habit plane have a tendency to move away from each other. Hence, close-packed 3D-structures are unlikely to be energetically favorable in Al-Cu alloys. According to our calculations the formation of platelets during the aging of Al-Cu alloys is in contrast to Al-Zn alloys, where obviously 3D-Clusters are preferred. Again, this is in accordance with experimental results for pure Al-Zn alloys, where clusters grow in early stages of decomposition in a spherical manner [41; 43; 45]. Conceptualization, P.F. and M.J.P. and T.E.M.S. ; Methodology, P.F. and M.J.P. ; Software, P.F. ; Validation, P.F. and M.J.P. and T.E.M.S. ; Formal Analysis, P.F. and T.E.M.S. ; Investigation, P.F. ; Resources, P.F. ; Data Curation, P.F. ; Writing Original Draft Preparation, P.F. and T.E.M.S. ; Writing Review & Editing, M.J.P. and I.W. ; Visualization, T.E.M.S. ; Supervision, M.J.P. ; Project Administration, M.J.P. and T.E.M.S. ; Funding Acquisition, P.F. and T.E.M.S. This research was funded in part by the German Research Foundation (Deutsche Forschungsgemeinschaft--DFG) grant number STA 527/3-2 and STA 527/5-1.