# The Correlation Factors and Mechanisms of Diffusion for P and S in the Cu Single Crystal
## 4 Discussion  
The value of \(D\) obtained for S was in the range of experimental and computational values for the diffusion of S in Ni. At 1000 \({}^{\circ}\)C, the value of \(D\) for the diffusion of S in Ni is 7.1 \(\times\) 10\({}^{-9}\) cm\({}^{2}\cdot\)s\({}^{-1}\)[7]; our data in Figure 4 showed that the corresponding value for S in Cu is 3.18 \(\times\) 10\({}^{-9}\) cm\({}^{2}\cdot\)s\({}^{-1}\). Both of the \(D\) values for S and P agreed with those previously published and with the fact that S is a much faster diffuser in Cu single crystals than P [45]. S is also a faster diffuser than P in polycrystalline Cu [41]. A polycrystalline material with a microstructure has extended defects such as GBs, where diffusion can be considerably faster than at the bulk. However, the rate-determining step for overall diffusion is largely determined by the diffusion at the bulk, provided that the impurities are homogeneously distributed in the materials at the start of the experiments. In experimental observations in polycrystals, S diffused 10 times as fast as P [41], while in the pure single crystal, the difference was of 20 times that at room temperature according to our data. The difference was even larger at high temperatures [45]. From the experimental data, it is known that the presence of GBs and other polycrystal defects reduces the differences in \(D\) values between S and P. The possible mechanistic origin for this observation can be as follows: the data presented in Table 4 show that the correlation factor for impurity diffusion, \(f_{2}\), was orders of magnitude smaller for S than for P. This is a limiting case where the stability of the vacancy-impurity complex is much higher for S than for P. The measures of this stability are, for each impurity, the following ratios: \(w_{3}/w_{0}\), which corresponds to the dissociation of the vacancy-impurity complex and was 1.79 times larger for P than for S, and \(w_{4}/w_{3}\), which corresponds to the probability of reassociation of the vacancy-impurity complex via \(w_{4}\) in case it was dissociated via \(w_{3}\), which gives a measure of the stability of the vacancy-impurity complex, and was 145 times larger for S than for P. Hence, the dissociation of the vacancy-impurity pair is much more unlikely to occur for S than for P. This means together with the magnitude of \(w_{3}\), which was much lower for S than for P, implies that the probability of the S-vacancy pair to diffuse as a single entity via the \(w_{1}\) and \(w_{2}\) jumps, forming a pair that is called a \"Johnson molecule\", is very high [46]. Additionally, because \(f_{2}\) is very close to zero for S, this also implies that this impurity diffusion is strongly correlated in the Cu single crystal, while P has a diffusion mechanism that is an almost ideal balance--50/50--between diffusion along a straight line and in random directions. However, the very low energy barrier for an impurity jump in the case of S, \(w_{2}\), led to a jump frequency which was six orders of magnitude larger than that of P (Table 3). This caused the diffusion of S to be faster than P despite being less efficient due to the correlation factor being close to zero. When extended defects, for example GBs, are present in the material, the diffusion of these impurities is largely constrained by the features of the defects that have well defined diffusion paths, such as channels [26, 47]. This implies that at these constrained diffusion pathways, random diffusion is largely constrained and both S and P will adopt similar diffusion mechanisms. This can cause a decrease in the large differences in the diffusion coefficients between S and P, because at the GBs, the differences in the correlation factor for their diffusion, \(f_{2}\), is smaller than that observed for the bulk single crystal. The origins of the effects that lead to differences in the \(D\) value and the correlation factors can be traced down to both geometric and electronic structure phenomena, as we recently detailed for the bonding of these impurities to Cu [26].