the coincidence-site lattice

Most grain boundaries are high-energy messes, but a few special misorientations are strangely well-behaved. Here is the idea. Take two identical crystal lattices, overlap them, and rotate one relative to the other. For most angles the two sets of points share almost nothing. But at certain magic angles, a regular fraction of the points of one lattice land exactly on top of points of the other. Those shared, overlapping points form their own repeating pattern: the coincidence-site lattice (CSL).

The CSL is described by a number, Sigma (the Greek letter), which counts how rare the coincidences are: Sigma is the ratio of the CSL unit-cell volume to the ordinary crystal unit-cell volume, so 1 in every Sigma lattice points is a coincidence site. A small Sigma means many shared sites, a well-matched, low-energy boundary. For example, in a cubic crystal a 36.9-degree rotation about a <100> axis gives Sigma = 5: one atom in five sits on a coincidence site. The perfect coherent twin in FCC is the extreme case, Sigma = 3. By convention Sigma is always an odd number.

A boundary that lies along a plane rich in coincidence sites can bond well across the seam and so has unusually LOW energy; these are the special boundaries. They resist sliding, migration, corrosion, and cracking better than random boundaries, which is exactly why grain-boundary engineering aims to pack a material with low-Sigma boundaries. Honest note: a low Sigma is necessary but not sufficient. The boundary must also lie on a good plane, and real boundaries only approximately hit the exact CSL angle, so the coincidence is usually a near-coincidence maintained by a few extra dislocations.