# Peierls energy

The maximum change in the total misfit energy is called the Peierls energy \(\Delta {U_P}\). It is the energy required to move unit length of dislocation over the resistance of the lattice. The following graph shows how the fraction of the unit cell across which the dislocation has moved affects the misfit energy. Adjusting parameters demonstrates how these affect the Peierls energy.

The misfit energy associated with in plane strains varies sinusoidally, as does the total energy. Whereas the misfit energy associated with the misalignments is greatest at the position of lowest overall energy. The period is b/2. Hence:

\[ \Delta {U_T}\left( \alpha \right) = \frac{1}{2}\Delta {U_P}\left( {1 - cos4\pi \alpha } \right) (1)\]

This is shown in the following interactive graph.

As can be seen in the magnified strain energy graph, the dislocation width also changes as the dislocation moves, although for simplicity simulations often assume it is constant.

The magnitude of the Peierl's energy (and hence the misfit energy) scales with G**b**^{2}. The Peierl's energy varies exponentially with w/b.

\[\frac{{{\rm{\Delta }}{{U}_{\rm{P}}}}}{{{G}{{\rm{b}}^2}}} ∝ \exp ( - \frac{w}{b}) (2)\]

As can be seen in the misfit energy graph animation, the dislocation width also changes as the dislocation moves, although for simplicity, simulations often assume it is constant. For example, in the energy calculation, we assumed that w/b was a constant for any value of α.