Statistics of vacancy-induced diffusion

One of the aspects associated with vacancy-mediated diffusion that differentiates it from the hopping mechanism on this surface is the long waiting time between consecutive jumps. An example of the distribution of waiting times has been plotted for both In and Pd in Fig. 5. 

As can be seen from the figure, both distributions are purely exponential. The exponential shape of the distributions shows that the waiting time of an embedded atom is governed by a Poisson process with rate t 1. This implies that subsequent long jumps are independent, which we take as proof 

The diffusion of the embedded atoms in the surface proceeds through multi-lattice-spacing jumps separated by long time intervals. The multi-lattice-spacing nature of the diffusion is illustrated by the jump length distributions which are plotted in Fig. 6. 

In Section 3 we derive that for the vacancy-mediated diffusion mechanism, one expects the shape of the jump length distribution to be that of a modified Bessel function of order zero. Both distributions can be fit very well with the modified Bessel function, again confirming the vacancy-mediated diffusion mechanism for both cases. The only free parameter used in the fits is the probability prec for vacancies to recombine at steps, between subsequent encounters with the same embedded atom [33]. This probability is directly related to the average terrace width and variations in this number can be ascribed to the proximity of steps. The effect of steps will be discussed in more detail in Section 4. 

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