## 3 Results  
### Reliability of Modeling  
To confirm the reliability of the employed numerical methods, we have calculated the formation energies (formation enthalpies at zero pressure) of mono- and di-vacancies in fcc Al. In the case of di-vacancies, we have considered nearest (1 NN) and next nearest neighbor (2 NN) configurations. The mono-vacancy formation energy is calculated as  
\[H_{\text{V}}^{\text{F}}=E_{\text{V(N-1)}}-\frac{N-1}{N}\,E_{\text{Al_{bulk}}}(N) \tag{1}\]  
where \(N\) is the number of atoms in the supercell, \(E_{\text{V(N-1)}}\) is the total energy of a fcc Al-supercell containing a mono-vacancy, and \(E_{\text{Al_{bulk}}}(N)\) is the total energy of a perfect fcc Al-supercell. The formation energies obtained are given in Table 1. Our VASP calculations for the isolated mono-vacancy lead to values in close agreement with previous similar LDA calculations by [PERSON] et al. [15] but also with a different approach like SIESTA [33] giving for the formation energy \(H_{\text{V}}^{\text{F}}=0.66\) eV [34; 35]. All the calculated values deviated less than 0.05 eV from reliable experimental values [16]. Here, reference [16] is a summary of a few dozen experimental works, where the data are weighted according to their relevance by experts in the field. So, the given value of \(H_{\text{V}}^{\text{F}}=0.67\) eV for the vacancy formation energy in pure aluminum is an average of the most reliable values published. Our results for the di-vacancy binding energies \(H_{\text{2V,X}}^{\text{F}}-2H_{\text{V}}^{\text{F}}\) (X = 1 NN or 2 NN) show that the interaction between nearest neighbor vacancies (X = 1 NN) in Al is repulsive. This is in agreement with the results by [PERSON] et al. [15] and also with the experimental finding that Al does not, in contrast to other metals like Cu [36; 37], show a tendency for forming vacancy clusters after low-temperature irradiation and subsequent annealing [38]. However, the 2 NN di-vacancy shows a tiny binding which is also in agreement with the results by [PERSON] et al. [15]. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Method** & **Volume Relax** & **MP-Mesh** & **Atoms** & \(\begin{array}{c}H_{\text{V}}^{\text{F}}\end{array}\) & \(H_{\text{LV,NN}}^{\text{F}}\) & \(H_{\text{2V,NN}}^{\text{F}}\) & \(H_{\text{2V,NN}}^{\text{F}}\) & \(H_{\text{2V,2 NN}}^{\text{F}}\) \\  & & & **(eV)** & **(eV)** & **(eV)** & **(eV)** & **(eV)** \\ \hline VASP-LDA & yes & \(6\times 6\times 6\) & 64 & 0.71 & — & — & — & — \\ VASP-LDA & no & \(6\times 6\times 6\) & 64 & 0.713 & 1.506 & 1.409 & +0.081 & \(-\)0.016 \\ VASP-LDA & no & \(6\times 6\times 6\) & 108 & 0.714 & 1.489 & 1.421 & +0.061 & \(-\)0.007 \\ VASP-GGA & no & \(6\times 6\times 6\) & 108 & 0.66 & — & — & — & — \\ SIESTA-DZP & no & \(3\times 3\times 3\) & 108 & 0.64 & — & — & — & — \\ Exp. [16] & — & — & — & 0.67 & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison with experimental results: Formation energies for mono- \(H_{\text{V}}^{\text{F}}\) and di-vacancies \(H_{\text{2V,X}}^{\text{F}}\) in the nearest neighbor (X = 1 NN) and next nearest neighbor (X = 2 NN) positions in fcc Al. The binding energies \(H_{\text{2V,X}}^{\text{b}}\) of the two vacancies in the two configurations are also given. Positive and negative binding energies indicate repulsion and binding, respectively. (SIESTA results: see [34]). To further check the reliability of our calculations in terms of supercell sizes and \(\mathbf{k}\)-point meshes, we have computed the solubility enthalpy of Cu in Al. It is calculated as  
\[\Delta H_{\text{mix}}=E_{\text{CuinAl}}-\left[\frac{N-1}{N}\,E_{\text{Albulk}}+ \frac{1}{N}\,E_{\text{Cubulk}}\right] \tag{2}\]  
where \(E_{\text{CuinAl}}\) is the total energy of an fcc Al-supercell containing one Cu-atom, while \(E_{\text{Albulk}}\) and \(E_{\text{Cubulk}}\) are the total energies of pure fcc Cu- and Al-supercells, respectively. Supercells of the same size are used to calculate these energies.