Twinning is crystallographically defined by its twin plane 𝑲, the mirror plane in the twin and parent material, and 𝜼 which is the twinning shear direction. Deformation twins in Zr are generally lenticular in shape, lengthening in the 𝜼 direction and thickening along the 𝑲 plane normal.
The twin plane, shear direction, and shear plane form the basis vectors of an orthogonal set. The axis-angle misorientation relationship between the parent and twin is a rotation of angle 𝜉 about the shear plane's normal direction 𝑷.
More generally, twinning can be described as a 180° rotation about an axis (𝑲 for type I twins or 𝜼 for type II twins normal direction) , or a mirror reflection in a plane (𝑲 or 𝜼 normal plane).
In addition to a homogeneous shear, atomic shuffles are sometimes required to reform the correct crystal structure in the twinned lattice. For each twin variant, a reciprocal twin with swapped 𝑲 and 𝑲 𝜼 and 𝜼 is possible, but one variant may appear more frequently in reality due to complexities with the required shuffles.
there are only two crystallographic planes in a shearing action that do not change their shape and size as a consequence of the shear. The first 𝑲 is the plane defining the upper and lower surfaces of the sheared volume. This plane contains the shear direction. The other plane, designated C. The shear direction is shown with an arrow and labelled with its customary designation 𝜼. It follows from the above that there are three ways that a crystal lattice can be sheared while still retaining its crystal structure and symmetry:
# When 𝑲 is a rational plane and 𝜼 a rational direction, a twin of the first kind
# When 𝑲 is a rational plane and 𝜼 a rational direction, a twin of the second kind, rare
# When all four elements 𝑲, 𝑲, 𝜼, and 𝜼 are rational, a compound twin