Collective cosine relations and the decay of the collective correlation process in solid-state diffusion

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Abstract
Certain relations between the collective ⟨cos θ(1) αβ⟩, where θ(1) αβ is the angle between the jump of some atom of the α species and the jth succeeding jump of any atom of the β species, are explored by means of a decay of the collective correlation process. It is found that the ⟨cosθ(1) αβ)⟩ can be expressed in terms of a power of the appropriate Lang;cos θ(1)⟩, the average collective cosine of the angle between consecutive jumps, should every atomic jump for the species lie along an axis of at least twofold symmetry and the collective correlations between consecutive jumps be independent of that between the previous jumps. In such circumstances, the correlation corresponds to an ‘ideal’ decay process with the single relaxation parameter being related simply and directly to ⟨cos(2)). However, study by computer simulation shows that these conditions are rarely met exactly for a random binary alloy except for ⟨cosθ(2)), which generally follows what is predicted by the ideal decay process.