Theory and Monte Carlo calculation of the tracer correlation factor

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Abstract
Some expressions for the tracer correlation factor f for diffusion by the vacancy mechanism are reviewed and their equivalence justified. In the case of the low-vacancy-concentration limit it is proved that the probability of the second jump of an atom being the reverse of the first is 2/Z where Z is the coordination number of the lattice. This result provides a simple derivation of f in some cases and gives a physical interpretation of the approximation f=1-2/Z. The Monte Carlo evaluation of f is discussed and quantitative estimates made of the number of atoms to be sampled to achieve a given accuracy. Calculated values of the averaged cosines of angles between the direction of an initial jump and the direction of the (k+1)th jump of an atom are presented for the cubic lattices and for the honeycomb lattice. It is deduced that the nature of the correlations in the low-vacancy-concentration limit are different to those at other vacancy concentrations.