the orientation relationship

When one crystal transforms into another — or a new phase grows inside a parent — the daughter does not usually appear at some random angle. It inherits a definite angular tie to the parent: specific planes and directions of the new crystal line up with specific planes and directions of the old one. That fixed angular link is the orientation relationship. It is like a photograph developing in a preferred alignment to the negative rather than at any random rotation.

It is stated by naming the planes and directions that stay parallel. For martensite in steel the Kurdjumov-Sachs relationship is {111}gamma parallel to {110}alpha-prime with <1-10>gamma parallel to <1-11>alpha-prime — the close-packed plane of the FCC parent lands on the close-packed plane of the BCC product, and a close-packed direction in each stays parallel. A closely related one is Nishiyama-Wassermann. These are not coincidences: they minimise the atomic misfit at the interface, so the boundary between parent and product is as coherent (well-matched) as possible.

The orientation relationship is a fingerprint of the transformation mechanism — you can read off whether atoms rearranged by a shear (displacive) or re-sorted by diffusion — and it is measured directly today by electron backscatter diffraction (EBSD) or selected-area electron diffraction. It also controls how many crystallographically equivalent product variants can form from one parent orientation (its symmetry-related copies), which sets the texture and internal structure of the transformed material.