# The Correlation Factors and Mechanisms of Diffusion for P and S in the Cu Single Crystal
## 2 Materials and Methods  
### The Effective Diffusion Model--5-Frequency Model  
An effective diffusion coefficient for a substitutional impurity must consider the correlations between the different jumps that can affect its mobility. The frequency (\(w_{i}\)) of each of these jumps (\(i\)) affects the probability of the impurity jump to a neighboring vacancy. The importance of the collection of \(w_{i}\) and the derivation of their expressions for the fcc lattice have been previously reported [5; 6]. The correlation for the diffusion of an impurity in the dilute limit, with no interactions between the impurities, is accounted for with the 5-frequency--or 5-jump--model, shown in Figure 1 for the fcc lattice. This model considers that these frequencies are the only vacancy jump frequencies in the crystal and it is thus a complete correlation model for the bulk of the fcc single crystal [6]. The nomenclature here used is the same as in the original papers [5; 6; 8] and of more recent work [10], where for clarity, all jump frequencies are denominated \(w_{i}\), and the index \(i=\) [0, 1, 2, 3, 4] represents the jump as in the original works. In this model, the diffusion coefficient for a substitutional impurity that diffuses via monovacancies in the dilute limit of a solid solution is  
\[D_{1}=D_{0}\,\frac{f_{2}}{f_{0}}\frac{w_{2}}{w_{0}}\frac{w_{4}}{w_{3}} \tag{3}\]  
where \(D_{1}\) is the diffusion coefficient of the monovacancy and \(D_{0}\) is the self-diffusion of the matrix atoms, \(f_{0}\) is the correlation factor for the diffusion of matrix atoms and \(f_{2}\) is the correlation factor for impurity diffusion. \(D_{0}=\) 7.161 \(\times\) 10\({}^{-37}\) cm\({}^{2}\cdot\)s\({}^{-1}\) at \(T=\) 298 K from a linear fit from previously published low temperature data [15]. At low defect and impurity concentrations [16], the correlation factor \(f_{0}\) for self-diffusion in the fcc lattice is 0.781 [5]. This factor accounts for the correlation effects between the vacancy and the diffusing Cu atom in the atomic jumps of Cu in the fcc lattice. The correlation factor \(f_{2}\) for the diffusion of a substitutional impurity by a monovacancy-assisted mechanism in a face-centered cubic lattice is given by [6; 8; 10]  
\[f_{2}=\frac{2w_{1}+7 Fw_{3}}{2w_{2}+2w_{1}+7 Fw_{3}} \tag{4}\]  
where \(F\) is the fraction of vacancies that make \(w_{3}\) jumps (Figure 1) that effectively do not return to the site from which the jump was made. This term is also called the fraction  
Figure 1: Vacancy jumps (\(w\)) considered in the 5-frequency model. These jumps account for all possible vacancy jumps in the fcc single crystal. Cu (\(\bullet\)), P or S impurity (\(\bullet\)), vacancy (\(\blacksquare\)). of dissociating vacancies. \(F\) has been derived for the fcc lattice [6; 17] with a basis in the Bardeen-Herring model [14] and is defined as  
\[F=1-\frac{y}{7}\left(\frac{11.56+50.10x+40.00x^{2}}{0.52+4.71x+12.06x^{2}+8.00x ^{3}}\right)\text{, with }y=\frac{w_{4}}{12w_{0}}\text{ and }x=\frac{(w_{4}-w_{0})}{12w_{0}} \tag{5}\]  
where \(F\) depends on the jump frequencies \(w_{4}\) and \(w_{0}\) shown in Figure 1. The jump frequencies \(w_{i}\) are given by  
\[w_{i}=v_{i}exp\left(\frac{\Delta S_{i}^{\ddagger}}{k_{B}}\right)exp\left(- \frac{\Delta H_{i}^{\ddagger}}{k_{B}T}\right) \tag{6}\]  
where \(v_{i}\) is the jump attempt frequency for the jump \(i\) as defined in Equation (1); this quantity is a vibrational frequency and its significance within the different transition state (TS) theories has been previously discussed in detail [10; 18]. It has been shown that \(v_{i}\) is very similar for the different jumps of the matrix atoms and the approximation \(v_{0}\approx v_{1}\approx v_{3}\approx v_{4}\) is valid without a significant loss of accuracy [6; 19]; \(\Delta S_{i}^{\ddagger}\) and \(\Delta H_{i}^{\ddagger}\) are the activation entropies and enthalpies for the jump \(i\) respectively and \(T\) is the temperature.