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In materials science, ring diffusion describes a class of atomic diffusion mechanisms by which atoms may diffuse through a material along a ring or loop of atoms. This mechanism was first proposed by Clarence Zener, who reported that self-diffusion along a loop of four atoms had a lower potential activation energy than diffusion between two sites (considered direct exchange). Zener’s initial characterization of a ring mechanism involved only the direct movement of atoms without vacancy.

Early mechanism theory
This mechanism utilizes the random walk model as a mathematical basis, describing a rotation of the ring by π/2 following each jump. Zener defined the experimental change in entropy from ring diffusion with the following:


 * $$ \frac{S}{k} = ln(\frac{D_0}{a^2v}) $$

where


 * a is the lattice parameter
 * D is the diffusion coefficient
 * v is the jump velocity

In cases of four atom rings in BCC and FCC lattices, the entropy change was found to be positive and the thermodynamic free energy contribution was found to be negative, indicating the feasibility of the mechanism.

Adjustments after Kirkendall discovery
Unlike similar mechanisms, ring diffusion is noted by confined movement where the diffusing atom does not migrate outside of the loop. Following the discovery of the Kirkendall effect, the ring mechanism received greater scrutiny since it could not account for the unequal migration rates of atoms from different species. However, despite being energetically unfavorable in comparison to newer models of diffusion – such as vacancy diffusion – evidence suggests that ring diffusion still occurs in specific instances. Updated ring diffusion mechanisms have combined the model proposed by Zener with vacancy transport to yield models that are consistent with the Kirkendall effect.

Two models noted by Tiwari and Mehotra are the triple-defect mechanism (TDM) and the six-jump vacancy mechanism (SJVM). Both mechanisms incorporate both vacancy movement and atomic relocation in a repeated sequence, similar to Zener’s original ring diffusion model. Additionally, both models have been observed and are energetically favorable, though not as likely as low-energy diffusion mechanisms like vacancy diffusion. In these models, vacancy movement is coordinated on a four or six-atom basis, matching the repeated region of migration in ring diffusion.

Occurrences in metals and metalloids
Gaurav Kumar identified another vacancy-facilitated form of the ring diffusion mechanism possible in metals. At temperatures below 900°C, experimental results identify that the diffusion coefficient in FCC metals deviates from what may be expected using the vacancy mechanism.

Kumar’s mixed mechanism identifies that at low temperatures, fewer vacancies are present in a metallic material. Therefore, ring diffusion is more likely, but the exact mechanism must utilize the remaining vacancies. The model implements experimental results using FCC copper and demonstrates that ring diffusion proceeds until atoms are aligned with a vacancy site. Since vacancy diffusion is highly favorable, the close atom will occupy the vacancy when it is near enough.

Another identified instance of a combined ring and vacancy diffusion mechanism occurs in the metalloid, silicon, where the introduction of a substitutional impurity may act as an attractive site that effectively pins a ring of migrating atoms. Because of the attraction to the impurity, silicon cycles around the site rather than diffusing freely. In particular, arsenic impurities are able to pin silicon atoms on the lattice within a 3-atom range. Like in metals, the domination of a mixed diffusion mechanism relies on temperature and the lower quantity of vacancies in certain ranges. Studies by Pankratov and Chesser note the implementation of ring diffusion and related phenomena at low to medium temperatures (below 1250°C in silicon). Understanding diffusion phenomena in metals and metalloids – particularly silicon – is critical for metallurgy and the development of electronics.