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Partial dislocation

## Summary

In materials science, a partial dislocation is a decomposed form of dislocation that occurs within a crystalline material. An extended dislocation is a dislocation that has dissociated into a pair of partial dislocations. The vector sum of the Burgers vectors of the partial dislocations is the Burgers vector of the extended dislocation.

## Reaction favorability

A dislocation will decompose into partial dislocations if the energy state of the sum of the partials is less than the energy state of the original dislocation. This is summarized by Frank's Energy Criterion:

{\displaystyle {\begin{aligned}|{\boldsymbol {b_{1}}}|^{2}>&|{\boldsymbol {b_{2}}}|^{2}+|{\boldsymbol {b_{3}}}|^{2}{\text{ (favorable, will decompose)}}\\|{\boldsymbol {b_{1}}}|^{2}<&|{\boldsymbol {b_{2}}}|^{2}+|{\boldsymbol {b_{3}}}|^{2}{\text{ (not favorable, will not decompose)}}\\|{\boldsymbol {b_{1}}}|^{2}=&|{\boldsymbol {b_{2}}}|^{2}+|{\boldsymbol {b_{3}}}|^{2}{\text{ (will remain in original state)}}\end{aligned}}}

## Shockley partial dislocations

Shockley partial dislocations generally refer to a pair of dislocations which can lead to the presence of stacking faults. This pair of partial dislocations can enable dislocation motion by allowing an alternate path for atomic motion.

{\displaystyle {\begin{aligned}{\boldsymbol {b_{1}}}\rightarrow {\boldsymbol {b_{2}}}+{\boldsymbol {b_{3}}}\end{aligned}}}

In FCC systems, an example of Shockley decomposition is:

{\displaystyle {\begin{aligned}{\frac {a}{2}}[10{\overline {1}}]\rightarrow {\frac {a}{6}}[2{\overline {1}}{\overline {1}}]+{\frac {a}{6}}[11{\overline {2}}]\end{aligned}}}

Which is energetically favorable:

{\displaystyle {\begin{aligned}|{\frac {a}{2}}{\sqrt {1^{2}+0^{2}+(-1)^{2}}}|^{2}>&|{\frac {a}{6}}{\sqrt {2^{2}+(-1)^{2}+(-1)^{2}}}|^{2}+|{\frac {a}{6}}{\sqrt {1^{2}+1^{2}+(-2)^{2}}}|^{2}\\{\frac {a^{2}}{2}}>&{\frac {a^{2}}{6}}+{\frac {a^{2}}{6}}\end{aligned}}}

The components of the Shockley Partials must add up to the original vector that is being decomposed:

{\displaystyle {\begin{aligned}{\frac {a}{2}}(1)=&{\frac {a}{6}}(2)+{\frac {a}{6}}(1)\\{\frac {a}{2}}(0)=&{\frac {a}{6}}(-1)+{\frac {a}{6}}(1)\\{\frac {a}{2}}(-1)=&{\frac {a}{6}}(-1)+{\frac {a}{6}}(-2)\end{aligned}}}

## Frank partial dislocations

Frank partial dislocations are sessile (immobile), but can move by diffusion of atoms.[1] In FCC systems, Frank partials are given by:

{\displaystyle {\begin{aligned}{\boldsymbol {b}}_{\text{frank}}=&{\frac {a}{3}}[{\text{1 1 1}}]\end{aligned}}}

## Thompson tetrahedron

Shockley partials and Frank partials can combine to form a Thompson tetrahedron, or a stacking fault tetrahedron.[2][3]

## Lomer–Cottrell lock

The Lomer–Cottrell lock is formed by partial dislocations and is sessile.[3][4]

## References

1. ^ Meyers and Chawla. (1999) Mechanical Behaviors of Materials. Prentice Hall, Inc. 217.
2. ^ "5.4.2 Dislocation Reactions Involving Partial Dislocations". Tf.uni-kiel.de. Retrieved 2013-09-21.
3. ^ a b W.D. Nix. "Partial Dislocation Tutorial for FCC Metals" (PDF). Imechanica.org. Retrieved 2013-09-21.
4. ^ Meyers and Chawla. (1999) Mechanical Behaviors of Materials. Prentice Hall, Inc. 218-219.