Theory of two-dimensional vacancy-induced tracer diffusion

THEORY OF TWO-DIMENSIONAL VACANCY-INDUCED TRACER DIFFUSION 3.1. Tracer diffusion on an infinite surface 

The problem of vacancy-mediated tracer diffusion in two dimensions has been investigated for a long time [40-44] and several different methods (simulation, analytical models, enumeration of trajectories, etc.) can be used to address it. The mathematics of this type of diffusion was solved first for the simplest case [41], when the diffusion of the vacancy is unbiased (all diffusion barriers are equal the tracer atom is identical to the other atoms), the lattice is two-dimensional and infinite. There is a single vacancy present that makes a nearest-neighbor move in a random direction at regular time intervals and has an infinite lifetime, as there are no traps. The solution is constructed by separating the motion of the tracer and that of the vacancy. The correlation between the moves of the tracer atom is calculated from the probability that the vacancy returns to the tracer from a direction, which is equal, perpendicular or opposite to its previous departure. The probability density distribution of the tracer atom spreads with 

The same problem has been solved in an alternate way for all dimensions [42]. From this solution one can calculate the number of tracer-vacancy exchanges up to time t. In two dimensions the distribution is geometric, with mean (log t)/tt. The continuum version of this problem has been considered as well in the form of an infinite-order perturbation theory [43] the solution matches the asymptotic form of the lattice model. 

In a very recent study the lattice calculations have been generalized to biased diffusion [44]. The difference between the tracer atom and the substrate atoms was taken into account by having different vacancy-tracer and vacancy-substrate exchange probabilities, while the rate of vacancy moves was kept constant. A repulsive interaction reduces, while a moderately attractive interaction increases the spreading of the tracer distribution. 

Although these exact solutions are closely related to the two systems that we discuss here, the differences, e.g. in boundary conditions and vacancy lifetime, make a direct comparison with experiments impossible. For this purpose we develop a model of tracer diffusion, which includes the essential properties of the experimental system. 

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