# 21.4: Dislocation Width

- Page ID
- 31769

For a given atomic configuration, we can work out the total energy and then try different values of w until we find the minimum. Practically (see graph below), this can be done by summing the energy over planes from n=-1000 to n=+1000 either side of the “extra plane of atoms”, since the effects of increasing the number of planes beyond this is negligible.

As w increases, the decrease in energy associated with localising the in-plane strains (which is why we said a dislocation should never form) is offset by the increase in misalignment energy, which increases with w. There is therefore a minimum in the misfit energy, which gives the dislocation width.

The following graph shows the sums of the misfit energies (in-plane, misalignment and total) resulting from the displacements of planes either side of the initial “extra half plane of atoms”. We can determine the dislocation width (w/b) for given parameters by considering the minimum of the total misfit energy curve.

If we look at the value of 0.25G**b**^{2} for the four materials given, it is similar to the energy value calculated by the model. This shows an approximate agreement with the linear elastic solution.

## What determines w/b?

- What we are interested in is the ratio of d/b. As d/b increases, w/b increases. d/b is determined by crystal structure.
**b**is the Burger's vector- d is the spacing of the close packed planes
- We also need to take into consideration partial dislocations. For example, copper is ccp so the slip system is {111} <110>. But the dislocations dissociate into two partial dislocations\(\frac{a}{6}\)<112>; hence the magnitude of b will be \(\frac{a}{\sqrt{6}}\) where a is the lattice parameter.
- Therefore, the ratio of d/b in copper will be \(\frac{a}{\sqrt{3}} \div \frac{a}{\sqrt{6}}=\sqrt{2}\)

- Poisson ratio (ν) affects the in-plane component of the misfit energy. A higher Poisson ratio means the width of the dislocation is greater.

## What determines the total energy?

- The total energy is affected by the magnitude of G
**b**^{2}, where G is the shear modulus and**b**is the Burger's vector.

The following clip shows the free surface of a Cd single crystal subject to tensile testing. Slip is occurring on a particular set of planes, and the set of ridges that form on the free surface are created by the arrival of sets of dislocations.