# 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
This was related with the lower \(\Delta E_{\text{a}}\) value that, in turn, depended on the most favorable bonding, due to the larger volume and the asymmetry in the impurity coordination to the vacancy that occurred for S. When the diffusion coefficients were determined from the whole set of data, according to Equations (3) to (6), we obtained the values given in Table 4. The temperature-dependent diffusion coefficients obtained from our data for S and P in Cu,  
Figure 3: Potential energy profiles for each of the jumps shown in Figure 1: data for the S 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 S or Cu atom jumped to. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Jump**} & \multicolumn{2}{c}{**P**} & \multicolumn{2}{c}{**S**} \\ \cline{2-5}  & \(\boldsymbol{\
u\) (Hz)} & \(\boldsymbol{w_{i}\) (Hz)} & \(\boldsymbol{\
u\) (Hz)} & \(\boldsymbol{w_{i}\) (Hz)} \\ \hline
1 & 8.982 \(\times\) 10\({}^{9}\) & 0.0905146 & 9.478 \(\times\) 10\({}^{9}\) & 0.0053790 \\
2 & 1.313 \(\times\) 10\({}^{10}\) & 0.0665103 & 9.031 \(\times\) 10\({}^{9}\) & 26661.4233049 \\
3 & 1.039 \(\times\) 10\({}^{10}\) & 9.5 \(\times\) 10\({}^{-6}\) & 1.048 \(\times\) 10\({}^{10}\) & 5.3 \(\times\) 10\({}^{-6}\) \\
4 & 8.701 \(\times\) 10\({}^{9}\) & 0.3974227 & 8.038 \(\times\) 10\({}^{9}\) & 32.2438087 \\
0 & 1.023 \(\times\) 10\({}^{10}\) & 1.020 \(\times\) 10\({}^{-4}\) & 1.023 \(\times\) 10\({}^{10}\) & 1.020 \(\times\) 10\({}^{-4}\) \\ \hline \end{tabular}
\end{table} 
Table 3: The values of \(\
u\) (Equation (8)) and \(w\) (Equation (10)) for each of the jumps (\(w_{i}\)) in the 5-frequency model, as shown in Figure 1, for P and S in Cu at 298 K.  
according to Equation (10), and the experimental data for the self-diffusion coefficient for Cu are shown in Figure 4.  
Figure 4: Arrhenius plots of the temperature-dependent self-diffusion coefficient \(D\) (cm\({}^{2}\)/s) for the single crystal Cu, obtained from experimental data (blue) [44]; and the impurity diffusion coefficients for P (orange) and S (grey) in Cu, estimated using the correlation factors calculated in this work. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Quantity**} & \multicolumn{2}{c}{**Impurity**} \\ \cline{2-3}  & **P** & **S** \\ \hline \multicolumn{3}{c}{\(T\) = 298 K} \\ \hline \(x\) & 324.53063 & 26,336.58531 \\ \(y\) & 324.61396 & 26,336.66865 \\ \(F\) & 0.28609 & 0.28572 \\ \(f_{2}\) & 0.57637 & 2.0195 \(\times\) 10\({}^{-7}\) \\ \(f_{0}\) (fcc lattice) & 0.781 & 0.781 \\ \(D\) (cm\({}^{2}\cdot\)s\({}^{-1}\)) & \(1.44\times 10^{-29}\) & \(2.95\times 10^{-28}\) \\ \(D\) (cm\({}^{2}\cdot\)year\({}^{-1}\)) & \(4.56\times 10^{-22}\) & \(9.30\times 10^{-21}\) \\ \(D\) (cm\({}^{2}\cdot\)100,000 year\({}^{-1}\)) & \(4.56\times 10^{-17}\) & \(9.30\times 10^{-16}\) \\ \multicolumn{3}{c}{_Diffusion length_} \\ \(\sqrt{(Dt)}\) & \(2.13\times 10^{-11}\) & \(9.64\times 10^{-11}\) \\ \multicolumn{3}{c}{_Diffusion length_} \\ \(\sqrt{(Dt)}\) & \(0.674\times 10^{-8}\) & \(3.05\times 10^{-8}\) \\ \((cm\cdot\)100,000 year\({}^{-1}\)) & \\ \multicolumn{3}{c}{_T = 373 K_} \\ \(D\) (cm\({}^{2}\cdot\)s\({}^{-1}\)) & \(6.61\times 10^{-24}\) & \(8.06\times 10^{-23}\) \\ \(D\) (cm\({}^{2}\cdot\)year\({}^{-1}\)) & \(2.09\times 10^{-16}\) & \(2.54\times 10^{-15}\) \\ \(D\) (cm\({}^{2}\cdot\)100,000 year\({}^{-1}\)) & \(2.09\times 10^{-11}\) & \(2.54\times 10^{-10}\) \\ \multicolumn{3}{c}{_Diffusion length_} \\ \(\sqrt{(Dt)}\) & \(1.44\times 10^{-08}\) & \(5.04\times 10^{-08}\) \\ \((cm\cdot\)year\({}^{-1}\)) & \\ \hline \multicolumn{3}{c}{_Diffusion length_} \\ \(\sqrt{(Dt)}\) & \(4.57\times 10^{-06}\) & \(1.60\times 10^{-05}\) \\ \((cm\cdot\)100,000 year\({}^{-1}\)) & \\ \hline \hline \end{tabular}
\end{table}