We start with two separate crystals. The lattices are identical with parameters b and d.
Initial atomic positions in the A plane are given by xA = nb
Initial atomic positions in the B plane are given by xB = nb ± b/2
The atoms in the plane shown in red in effect become an edge dislocation when the crystals are joined.
On joining the crystals to form a dislocation, the atoms relax to minimise the overall misfit energy
Misfit energy arises from:
a) Strains caused by displacements Δx within the A, B planesTo determine the final atomic positions we can consider only the A and B plane atoms.
Here the atoms either side of the dislocation line have been numbered.
The final atomic positions for the A/B planes xA' and xB' are given by xA + uA and xB + uB respectively, where u represents the displacement from original position
The displacements (shown in red) are given by uA and uB
The displacements are symmetrical either side of the displacement line
Far from the dislocation, the atoms are aligned such that:
The displacement u(x) varies as arctan(x) with distance from the dislocation line
k and w are constants to be determined
The displacement u(x) varies as arctan(x) with distance from the dislocation line
k and w are constants to be determined
To determine k we consider the values of u at x=0 and x=∞
At x=∞, tan-1(x/w) = π/2 and uA = -b/4
→ k = b/2π
w is the dislocation width. To find it we need to minimise the overall misfit energy. This is explained in the next animation.