# The Correlation Factors and Mechanisms of Diffusion for P and S in the Cu Single Crystal
## 3 Results
### Vacancy Formation Energies and Vacancy-Impurity Interactions
The corresponding jumps are illustrated in Figure 1. Jump \(w_{0}\) was independent of the impurities, but it is shown in both figures for comparison; all remaining jumps were impurity specific. The DFT data used to determine the correlation parameters for the 5-frequency model are given in Table 2.  
The jump of S to the vacancy, \(w_{2}\), had less than half of the \(\Delta E_{a}\) of the corresponding jump of P. The jump of the Cu atom away from the impurity, \(w_{4}\), also had a considerably lower \(\Delta E_{a}\) for S than for P. The remaining Cu atom jumps that could be affected by the type of impurity--\(w_{3}\) and \(w_{1}\)--had similar \(\Delta E_{a}\) values for both P and S. Because \(w_{3}\) and \(w_{1}\) are similar, the considerably lower \(\Delta E_{a}\) for \(w_{2}\) and \(w_{4}\) for S when compared to P agreed with the experimental observation that S is a faster diffuser than P in Cu over a wide range of temperatures [41]. It is known that substitutional P is a fast diffuser in Ni via a vacancy-mediated mechanism [42; 43]. Its diffusion coefficient is highly dependent on temperature. In Cu, this was also expected as the mechanisms of impurity diffusion and segregation share many commonalities with Ni. The data in Table 2 show additionally that the magnitude of the ZPE correction was at most 0.014 eV. The differences between the corresponding ZPE for the jumps of S and P were at most 0.007 eV. We can thus conclude that the inclusion of the computationally demanding ZPE was not necessary to obtain accurate data on the diffusion of these impurities. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**P**} \\ \hline Jump & \(\lambda\) (Å) & \(\Delta\)E\({}_{\text{a}}\) (eV) & \(\Delta\)ZPE (eV) & \(\Delta\)\(H_{\text{a}}\) (eV) \\ \hline \(w_{1}\) & 2.518 & 0.639 & 0.012 & 0.650 \\ \(w_{2}\) & 2.501 & 0.663 & 0.005 & 0.668 \\ \(w_{3}\) & 2.545 & 0.884 & 0.005 & 0.889 \\ \(w_{4}\) & 2.521 & 0.626 & \(-\)0.014 & 0.611 \\ \(w_{0}\) & 2.495 & 0.819 & 0.008 & 0.828 \\ \hline \multicolumn{4}{c}{**S**} \\ \hline Jump & \(\lambda\) (Å) & \(\Delta\)E\({}_{\text{a}}\) (eV) & \(\Delta\)ZPE (eV) & \(\Delta\)\(H_{\text{a}}\) (eV) \\ \hline \(w_{1}\) & 2.518 & 0.714 & 0.010 & 0.724 \\ \(w_{2}\) & 2.501 & 0.315 & 0.012 & 0.327 \\ \(w_{3}\) & 2.545 & 0.895 & 0.009 & 0.904 \\ \(w_{4}\) & 2.459 & 0.505 & \(-\)0.008 & 0.496 \\ \(w_{0}\) & 2.495 & 0.819 & 0.008 & 0.828 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Distance of the jump (\(\lambda\)), activation energy (\(\Delta\)E\({}_{\text{a}}\)), contribution of the zero-point vibrational energy at the transition state (\(\Delta\)ZPE) and activation enthalpy (\(\Delta\)\(H_{\text{a}}\)) values for the corresponding jumps for the Cu atoms and the P and S that are represented in Figure 1.  
Figure 2: Potential energy profiles for each of the jumps shown in Figure 1: data for the P atom. The minimum at zero is at a substitutional lattice site and the other minimum corresponds to another substitutional lattice site initially occupied by a vacancy to which the P or Cu atom jumped to. The quantities necessary to obtain the correlation factors that accounted for the effective diffusion coefficients of P and S in Cu, as described in Equations (3) to (10), are given in Table 3.  
The fastest diffusion of S when compared to P had most of its origins in the \(w_{2}\) frequency, which was five orders of magnitude larger for S, as shown in Table 3. There was also a considerable contribution due to the three orders of magnitude differences in \(w_{4}\).