a tilt boundary

There are two clean ways to misorient one grain from its neighbour, and a tilt boundary is the first. Picture two identical books of stacked pages. Keep the boundary plane fixed like a wall between them, and rotate one book about an axis that lies IN that wall, so the axis is parallel to the boundary. The pages (atomic planes) now fan out at a small angle to each other, as if the two crystals were tilted toward one another. That kind of boundary is a tilt boundary.

A tilt boundary is defined by a rotation axis that lies WITHIN the boundary plane. For a low-angle tilt boundary the misfit is beautifully simple: it is accommodated by a vertical wall of edge dislocations, stacked one above the other, all with the same Burgers vector. The tilt angle theta and the dislocation spacing D are linked by D = b / (2 sin(theta/2)), about b/theta for small angles, so more tilt means more closely spaced edge dislocations. Each extra half-plane wedges the two crystals apart by just the right amount to make up the tilt.

The tilt boundary is the textbook proof that a grain boundary is built from dislocations, and it is the simplest boundary to model and to see. Contrast it with a twist boundary, where the rotation axis is perpendicular to the boundary plane and the misfit is carried by a crosshatched grid of screw dislocations instead. Honest note: real boundaries are usually mixed, part tilt and part twist, since a general misorientation axis lies neither exactly in nor exactly perpendicular to the boundary plane.