# The Correlation Factors and Mechanisms of Diffusion for P and S in the Cu Single Crystal
## 1 Introduction  
In the Swedish repository for spent nuclear fuel (KBS-3), the fuel is encapsulated in a cylindrical iron container that, in turn, is protected from the environment with an outer copper cannister [1]. The copper cannister serves thus as the ultimate barrier that separates the iron container from the environment. The copper barrier is expected to keep the iron container and the fuel protected from the environment for at least 100,000 years. Within this time frame the mobility of impurities that can eventually affect the properties of the copper material has to be well understood for proper safety assessment. A complete description of impurity diffusion in a lattice needs to account for the correlation between the movement of the impurity and the movement of the surrounding lattice atoms [2]. The quantity that accounts for this is the correlation factor and it is defined as the ratio of the mean squared displacement of the impurity to the mean squared displacement of a random walker in the same lattice [3]. Within random walk theory [4], thecorrelation factor quantifies how much the directions of the successive impurity jumps are random. The correlation factor is affected by the binding energy between the impurity and the surrounding lattice atoms as well as aspects related to the geometry of the bonding that will affect the transition states for the diffusion jumps [2; 5; 6]. For the case of substitutional impurities, whose diffusion mechanism is via vacancies, the binding energy between the impurity and the vacancy has important implications, both in the diffusion mechanism and in the diffusion rate. The binding energy of the impurity to the vacancy is directly proportional to the diffusion coefficient but it also depends on the correlation factors between matrix atoms and impurities [6; 7; 8]. Thus, correlation terms for the diffusion of vacancies, for the diffusion of the matrix atoms--self-diffusion--and of the impurities are necessary to account for the complete correlation picture that determines the migration of substitutional impurities, and that affects the directions of the successive atomic jumps and consequently the diffusion coefficient [2]. The diffusion coefficient \(D\) for isotropic diffusion is given by  
\[D=\frac{1}{6}\lambda^{2}\
u \tag{1}\]  
where \(\lambda\) is the jump distance and \(\
u\) is the jump frequency. This expression would be complete if the direction of the diffusive jump did not depend on the directions of the previous jumps. However, because of the correlated mobility, the direction of the jump depends to some degree on the directions of previous jumps [6] and the expression for \(D\) has to be modified to account for these correlations, and thus it becomes  
\[D=\frac{1}{6}\lambda^{2}\
u f \tag{2}\]  
where \(f\) is the correlation factor that considers the degree of correlation between the successive jumps. A widespread convention used for defining the correlation factor is the one previously proposed by [PERSON] [9]. In this model, correlation factors for a cubic lattice have values in the interval [0, 1] refs. [2; 4], proportionally affect the diffusion coefficient and depend on the crystal structure [2; 5]. If \(f\) is close to 0, it implies that the impurity moves with a high degree of correlation with the previous jumps, while the closer \(f\) is to 1, the more randomly the impurity moves. The degree of correlation is determined by the chemical bonding between the matrix and the impurity, and \(f\) is also affected by the effects that vacancy mobility has on the atomic jumps of both the matrix and impurity atoms. The correlation factors for self-diffusion in the matrixes of different structural ordering are known and their derivation has been previously detailed [5]. But the correlation factors for interactions between impurities and different matrixes must be determined for each case, considering the possible mechanisms for vacancy jumps [6], because both matrix atoms and impurity diffusion depend on the vacancy jumps.