Correlation of Atomic Jumps

We have seen that the elementary steps of atomic particles in crystals are normally correlated. In one way or the other, particles have not lost all memory about a previous step when they go on to the next one. This point was already addressed in Sec- 

Correlation diminishes the effectiveness of atomic jumps in diffusional random motion. For example, when an atom has just moved through site exchange with a vacancy, the probability of reversing this jump is much higher than that of making a further vacancy exchange step in one of the other possible jump directions. Indeed, if z is the coordination number of equivalent atoms in the lattice, the fraction of ineffective jumps is approximately 2/z (for sufficiently diluted vacancies as carriers) [C. A. Sholl (1992)]. 

This fraction is determined by the step-dance between a specified vacancy and the (tagged) atom during their encounter, which does not end before the atom-vacancy pair has definitely separated. Normally, a new and independently moving vacancy comes along much later and begins the next encounter with the tagged atom. 

The mean square displacement R2 of a particle after correlated or uncorrelated jumps is given as the mean vector sum 

Up to this point we have assumed implicitly that each defect responsible for the atomic motion has an infinite lifetime. In real crystals, however, this lifetime is finite because of the dynamic nature of the point defect equilibria. This means that only m consecutive jumps are correlated (corresponding to the defect lifetime). It has been shown [R. Kutner (1985)] that under these conditions 

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