Crystallography · Interfaces series
Interfaces & the Coincidence Site Lattice
Grain boundaries have deep geometry lore, and Sigma 3 is the fan favorite.
How two crystal lattices meeting at an angle settle into a repeating pattern of good and bad fit, and how that pattern sets the spacing of the dislocations that hold the boundary together.
Almost every polycrystalline material is mostly interface once you zoom in far enough: grain boundaries, phase boundaries, twin boundaries. None of them are beyond geometry: even a so-called random high-angle boundary has an exact description. Two lattices meeting at a boundary can always be described by a rotation (and, for a phase boundary, a change of lattice parameter too), and that geometric description turns out to predict a great deal about the boundary's structure, energy, and the dislocations needed to hold it together. This lab works through that geometry in four connected steps: the simplest case first (misfit and tilt boundaries), then the special angles where the two lattices share a repeating set of points exactly (the coincidence site lattice), then the general construction that works at any angle (the O-lattice), and finally what happens to a near-perfect coincidence boundary when the angle is off by a little (secondary dislocations and the DSC lattice).
Every number on this page is computed live from the same verified geometry, not looked up in a table. Pick an axis, drag an angle, and watch the two lattices and the count of coincident points change together.
Misfit & tilt boundaries
Shown as a genuine 3D slab of each crystal meeting at a planar interface, not a flat cross-section: an edge dislocation's "extra half-plane" is drawn as an actual plane of atoms, and the dislocation itself as the line where that plane terminates at the interface: a real 3D dislocation is a line, which a 2D cross-section can only ever show as a single point. Drag to orbit, scroll/pinch to zoom.
What's going on
A small-angle tilt boundary is just a wall of edge dislocations. Each dislocation supplies one extra lattice plane on one side, and the spacing between them sets the average misorientation. Frank's formula makes this exact for the geometry of a symmetric tilt boundary: D = b / (2 sin(θ/2)), which collapses to the familiar D ≈ b/θ once θ is small and expressed in radians. The same bookkeeping applies to a coherency-relieving misfit boundary between two phases with different lattice parameters, except the driving quantity is the misfit δ instead of a rotation angle, and the spacing is D = b/δ. Neither formula is exact once the dislocation cores start to overlap, which is why both are drawn here as a good approximation for small angles and misfits rather than a universal law: past roughly 10–15°, or a few percent misfit, the boundary is better thought of as a distinct structural unit than as a countable set of isolated dislocations.
Coincidence site lattices
Shown below as a genuine rotatable 3D sample of the lattice, not flattened to a 2D projection: coincidence is tested in the full 3D lattice, since testing a single atomic plane in isolation can be misleadingly over- or under-coincident (a subtlety this lab's own build process ran into and corrected). Drag to orbit and see the coincidence pattern from any angle, or jump to the classic view straight down the rotation axis.
| Σ | θ |
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Drag to orbit · scroll/pinch to zoom.
Why only special angles coincide exactly
Rotate one lattice against an identical copy of itself and, almost always, no point of the rotated lattice will land exactly back on a point of the original: the two are mutually irrational. At a discrete set of special angles, though, a fraction 1/Σ of the rotated lattice's points land exactly on original-lattice points, building up a coarser, exactly-repeating coincidence site lattice with a unit cell Σ times larger in area than the crystal's own cell. Ranganathan's 1966 relation generates every such angle for a cubic crystal: for a rotation axis [uvw] with N = u²+v²+w², and any pair of coprime integers m, n, the rotation θ = 2 arctan(n√N/m) generates a CSL with Σ equal to m²+n²N, divided by two as many times as needed to leave an odd number. By convention Σ is always reported as odd. Smaller Σ means a denser, more tightly matching coincidence pattern; boundaries at or near low-Σ misorientations are common in real microstructures (annealing twins are the Σ3 case) precisely because that geometric coincidence tends to correlate, imperfectly, with a lower-energy, better-fitting interface.
The same boundary, many labels
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The O-lattice
Rotation is about a fixed [110] axis (matching the snap-to-CSL button), mapped to world "up." At zero misfit the O-lattice degenerates from points to continuous rows parallel to the axis (shown as lines), since a pure rotation leaves the axis direction itself unchanged and there is nothing for it to reconcile there without a misfit; any nonzero misfit breaks this into a genuine 3D point lattice. Drag to orbit, scroll/pinch to zoom.
The general construction
Coincidence site lattice theory only answers the question at a handful of exact angles. Bollmann's O-lattice generalizes it to any rotation and any misfit at all, by treating each crystal as a continuum reference frame rather than a strictly discrete set of points. If A is the transformation (rotation, and dilation for misfit) that carries crystal 1 into crystal 2, the O-lattice points satisfy xO = (I − A−1)−1 b for every lattice translation vector b of crystal 1. Each O-point marks a place where the local relationship between the two lattices repeats, in the sense that best matches a rigid displacement, not a place where atoms literally coincide (that only happens at exact CSL angles with zero misfit). Physically, the spacing between O-lattice points is the spacing you would need between interface dislocations to relieve the rest of the misfit elastically and leave the O-points themselves in good register, which is exactly Frank's formula generalized beyond the small-angle, single-parameter case: at zero misfit the O-lattice spacing converges on b/(2 sin(θ/2)) as θ shrinks, and at zero rotation it converges on the misfit spacing b/δ, but away from those limits (rotation and misfit at once, which is the normal case for a real phase or grain boundary) the two simple formulas don't just add, and the exact O-lattice construction is what actually tracks the geometry.
Secondary dislocations & the DSC lattice
CSL and DSC cells are shown as genuine nested 3D cells (volumes Σ × and ÷ the unit cell, not just areas), with secondary dislocations drawn as the lines they physically are, running along the boundary. Drag to orbit, scroll/pinch to zoom.
The lattice that's coarser and finer
The DSC lattice (displacement shift complete lattice) is the coarsest lattice that still contains every translation vector of both crystals when they're expressed in one shared frame. It is, in a precise sense, the geometric complement of the CSL: where the CSL is a coarser lattice than the crystal's own (its cell is Σ times larger in area), the DSC lattice is a finer one (its cell is Σ times smaller). That reciprocal relationship is exact: (area of CSL cell) × (area of DSC cell) = (area of the ordinary unit cell)².
The DSC lattice matters because its vectors are the only Burgers vectors a dislocation confined to the boundary can have without destroying the coincidence pattern. When the actual misorientation misses an exact CSL angle by a small amount Δθ, that deviation is taken up by a periodic array of these secondary (or intrinsic) grain boundary dislocations, with Burgers vector equal to the shortest DSC lattice vector and spacing given by the same Frank-type bookkeeping as before, D = bDSC/Δθ, just with the much smaller DSC Burgers vector standing in for the full lattice vector. Because bDSC shrinks as 1/Σ, low-Σ boundaries can tolerate more angular deviation before their secondary dislocation spacing becomes unphysically small (or the boundary simply relaxes to a different, nearby low-Σ misorientation instead).
Where this model breaks down
Everything above is a geometric theory: it describes how two rigid, undistorted lattices line up, and infers dislocation content from that geometry alone. Real boundaries don't stay rigid. Atoms near the interface relax off their ideal geometric sites to lower the local energy, sometimes substantially, which is exactly why atomistic and DFT studies of grain boundaries routinely find structures that geometric CSL/O-lattice theory alone would not predict. Elastic anisotropy (real crystals are not elastically isotropic, and grain boundary dislocation energies and preferred spacings depend on it) is left out of Frank's formula entirely. Real boundaries also facet onto low-energy planes rather than staying on the flat plane a simple construction assumes, and the linear-elastic picture of isolated, non-interacting dislocations itself only holds while their cores stay well separated, which is precisely the small-angle, small-deviation regime this lab keeps its sliders in. Perhaps the most important caution: coincidence, by itself, is a weaker predictor of boundary energy than the original CSL picture assumed. Modern boundary-energy studies (misorientation and inclination scans across the full five-parameter space, not just the three describing misorientation) find plenty of low-energy boundaries far from any low-Σ coincidence, and plenty of high-Σ boundaries that are unremarkable. CSL and O-lattice theory remain genuinely useful for organizing and predicting dislocation content at a boundary of known misorientation; they are not, on their own, a reliable route to which boundary a material will actually form.
References
- Ranganathan, S. "On the geometry of coincidence-site lattices." Acta Crystallographica 21, 197–199 (1966).
- Bollmann, W. Crystal Defects and Crystalline Interfaces. Springer, 1970.
- Frank, F. C. "The resultant content of dislocations in an arbitrary intercrystalline boundary." In Symposium on the Plastic Deformation of Crystalline Solids, Carnegie Institute of Technology, 1950.
- Bhadeshia, H. K. D. H. Worked Examples in the Geometry of Crystals, 2nd ed. Institute of Materials, 2001. (Topic outline for this series; equations and worked examples above are derived independently from the standard literature, not reproduced from this text.)
BibTeX
@misc{tripathy2026interfacescsllab,
author = {Tripathy, Manisha},
title = {Interfaces & the Coincidence Site Lattice},
year = {2026},
howpublished = {\url{https://untetheredatom.com/crystallography/interfaces-csl-lab}},
note = {Interactive teaching resource}
}