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.
Two crystals meeting at a boundary are described by a rotation, plus a change of lattice spacing if they are different phases. That geometry predicts which dislocations the boundary needs. The six tabs go from the simplest case to the full picture: tilt and misfit boundaries, coincidence site lattices, the O-lattice, secondary dislocations, the boundary plane, and how boundaries move. Every picture and number is computed live from the lattices.
Misfit & tilt boundaries
Tilt one crystal against the other and the boundary needs a row of edge dislocations. Drag θ, or switch to Misfit and move the two lattice parameters, and watch the dislocations crowd together or spread apart.
In a tilt boundary the atom size does not change the picture: the number of atoms between dislocations depends only on θ. Bigger atoms would only zoom the same picture, so the metal just converts D to nm.
A tilt boundary is a wall of dislocations
Each dislocation adds one extra row of atoms on one side. They sit D = b / (2 sin(θ/2)) apart, about b/θ for small θ (Frank, 1950).
A misfit boundary works the same way
When a₁ and a₂ differ, the columns fall out of step. The crystal with the smaller spacing gets an extra column every a₁a₂/|a₂ − a₁|, about b/δ.
Where the picture stops working
Past about 10 to 15° of tilt, or about 10% misfit, the dislocation cores overlap. The boundary is then better described by repeating structural units than by separate dislocations.
More detail: how the picture is built
Both crystals are simple square lattices seen along the dislocation lines. For tilt, the top crystal is turned +θ/2 and the bottom one −θ/2. The rows that run almost parallel to the boundary end on it; each end is one extra half-plane. The bottom crystal is shifted by half a row so the dislocations from the two sides alternate, which gives exactly Frank's spacing.
Atoms are drawn at their ideal positions. In a real boundary they relax around each dislocation core, so the cores are spread over a few atoms, not one point.
Coincidence site lattices
Turn crystal 2 about the chosen axis. Only at a few special angles do lattice points of the two crystals land exactly on each other. Drag the slider slowly, or click a row in the table.
Drag slowly: rings appear only at the exact angles in the table (the slider snaps within 0.15°). Click a row to jump there.
| Σ | θ |
|---|
Seen straight down the axis, so each dot is a column of atoms. A ring means that column holds a point where both crystals have an atom, tested in 3D. Along [111] every column of both crystals overlaps at 60°, yet only 1 in 3 holds a shared point: the atoms in the other columns sit at different heights.
The same boundary, many labels
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Coincidence sites
At a CSL angle, 1 in Σ lattice points belongs to both crystals. These shared points form a larger lattice, the coincidence site lattice (the green cell is one of its cells).
Which angles work
For axis [uvw], N = u²+v²+w², and whole numbers m, n with no common factor: θ = 2 arctan(n√N/m) and Σ = m² + n²N, halved until odd (Ranganathan, 1966).
Why Σ matters
Low-Σ boundaries, such as Σ3 annealing twins, are common in real metals. A good geometric fit often goes with low energy, but not always: Mode 5 shows one Σ spanning a twentyfold range.
The O-lattice
Lay crystal 2 on top of crystal 1 at any angle and any misfit. The patches where the two lattices match are centred on O-points. Use the three example buttons to compare pure rotation, pure misfit, and both together.
O-points
xO = (I − A−1)−1 b for every lattice vector b, where A turns and stretches crystal 1 into crystal 2 (Bollmann, 1970). O-points are where the lattices are locally in step, not places where atoms sit on top of each other.
The dislocations go between them
The O-point spacing is the spacing the boundary dislocations need. Pure rotation gives Frank's b/(2 sin(θ/2)); pure misfit gives (1 + ε)/ε.
Rotation and misfit together
The two simple formulas do not add. With both present the O-cell tilts and changes shape (watch the green cell), and only the O-lattice gets the spacing and direction right.
Look for the patches where blue and red dots sit on top of each other: each patch is centred on an O-point. Between patches the two lattices are out of step, and that is where boundary dislocations go.
Secondary dislocations & the DSC lattice
Start at an exact CSL angle, then turn a little further. Drag Δθ and watch the good-fit patches shrink and the secondary dislocations crowd together.
Drag the slider and watch the boundary strip below: more deviation means more secondary dislocations, packed closer together.
The DSC lattice
The finest lattice that contains every lattice vector of both crystals. Its vectors are the Burgers vectors a boundary dislocation can have without spoiling the CSL fit on either side.
Secondary dislocations
A small extra turn Δθ is taken up by dislocations with b = bDSC, spaced D = bDSC / (2 sin(Δθ/2)), about bDSC/Δθ.
Brandon's limit
bDSC is short, so D shrinks quickly as Δθ grows. Past about 15°/√Σ the cores overlap and the boundary is no longer counted as that Σ (Brandon, 1966).
They can carry steps
A boundary dislocation can also shift the boundary plane by a step. Then it is called a disconnection (Hirth and Pond, 1996). Mode 6 shows how these move a boundary.
At the exact CSL angle the shared sites repeat evenly across the whole picture. Move off it and they survive only in patches. The boundary keeps the good CSL fit inside each patch and puts one secondary dislocation between patches, so the patch spacing is the dislocation spacing D.
The boundary seen face-on, 60 a wide
More detail: CSL and DSC sizes, and twin disconnections
The CSL cell holds Σ ordinary cells; the DSC cell is Σ times smaller. So (volume of CSL cell) × (volume of DSC cell) = (volume of ordinary cell)².
bDSC here is the shortest DSC vector lying in the picture plane, found from all differences between a crystal-1 and a crystal-2 lattice vector. The picture uses a simple cubic lattice.
At a Σ3 coherent twin two disconnections dominate: a pure step (h = 3 layers, b = 0) and a twinning step (h = 1 layer, b = a0/6⟨112⟩). This tab does not fix a boundary plane, so it shows the dislocations without steps.
Boundary plane & the Σ3 family
Same two crystals, same twin relation (60° about ⟨111⟩). Only the boundary plane changes. Drag the slider, or drag on the picture, and watch the energy change by a factor of twenty.
Energy vs. plane tilt
0°: coherent twin (CTB)
The boundary lies on {111}. The two crystals are mirror images, and every boundary atom sits where both crystals want it. Copper: about 22 mJ/m² (EAM calculation).
90°: incoherent twin (ITB)
The boundary lies on (112), at right angles to {111}. Atoms across it do not line up, and the misfit repeats every three {111} layers. Copper: about 460 mJ/m².
In between: steps
In copper and aluminium simulations the boundary splits into CTB and ITB steps (Tschopp and McDowell, 2007). The energy is the mix: E = ECTB cos φ + EITB sin φ, rising smoothly to about the ITB value.
What EBSD sees
Every boundary on the slider has the same misorientation, so EBSD labels all of them Σ3. Only the boundary plane separates a 22 mJ/m² twin from a 460 mJ/m² one. Tick “EBSD view” to see this.
More detail: which planes, stacking faults, Brandon’s rule, boundary engineering
Which planes. The slider tilts the boundary about [110]. In that zone the plane at 90° from (111) is (112), the incoherent twin. The plane at 19.47° is (112); it mirrors to (552) in the other crystal, so it is an asymmetric boundary, not the ITB. (110) sits at 35.26° and (001) at 54.74°; the energy has no dip at either.
Why only 3 of 5 numbers is not enough. Modes 1 to 4 describe the misorientation: rotation axis and angle, three numbers. A grain boundary needs two more, the direction of the boundary plane. For Σ3 those two numbers change the energy more than twentyfold.
Twin energy and stacking fault energy. In the simplest stacking model, which counts only neighbouring close-packed layers, a twin boundary costs about half the intrinsic stacking fault energy (γISF ≈ 2γtwin). CTB values used here: Cu 22, Al 75, Ni 64, Ag 8 mJ/m²; ITB values: Cu 460, Al 320, Ni 530, Ag 280 mJ/m² (EAM calculations: Tschopp and McDowell, 2007; Olmsted, Foiles and Holm, 2009; Tschopp and Coleman, 2015). The step model uses only these two end values per metal.
Brandon’s rule. EBSD software calls a boundary Σ3 if it is within 15°/√3 ≈ 8.7° of the exact twin misorientation (Brandon, 1966). That window says nothing about the plane, so a Σ3 fraction from a 2D map mixes coherent twins with higher-energy Σ3 boundaries. Five-parameter analysis, which adds the plane from serial sections or stereology, separates them.
Boundary engineering. Low-energy boundary planes are the most common ones in annealed metals (Rohrer, 2011), so most Σ3 boundaries in annealed FCC metals lie on {111}. When grain boundary engineering (Watanabe, 1984) raises the Σ3 fraction, the benefit comes mainly from these coherent twins, not from the Σ3 misorientation as such.
How boundaries move: disconnections
A boundary moves one step at a time. Send a step along the coherent twin and watch the boundary rise. Follow the black marker line to see whether the crystal above also slides sideways.
Drag the slider to 100%, or drag across the picture, to finish a step.
What one step does
Boundaries move by steps
A boundary does not move as a flat sheet. It moves when line defects called disconnections run along it. Each one has a step height h and a Burgers vector b (Hirth and Pond, 1996).
Twinning step: rise and slide
b = a0/6⟨112⟩, h = one {111} layer. Each pass moves the twin boundary up one layer and slides the crystal above by b. Because it has a b, shear stress pushes it.
Shear coupling, β = b/h
For the twinning step b/h = 1/√2 ≈ 0.71, the twinning shear of FCC metals. Many grain boundaries show this link between moving and shearing (Cahn, Mishin and Suzuki, 2006).
Pure step: rise, no slide
b = 0, h = three layers. It moves the boundary without changing the crystal's shape, so shear stress cannot push it. A difference in energy between the two grains can, such as stored energy from deformation.
More detail: where b and h come from, and what the picture leaves out
Where b and h come from. The Burgers vector of a disconnection must be a vector of the DSC lattice from Mode 4, so that the boundary structure is the same on both sides of it. The step height must be one the boundary can actually have. For a given boundary there are many allowed (b, h) pairs; each gives its own coupling factor β = b/h (Han, Thomas and Srolovitz, 2018).
Why the pure step shuffles. Inside a 3-layer pure step the first layer moves by one Shockley partial and the second by another, and the three partials of the step add up to zero. So the layers change from crystal 1 stacking to twin stacking, but the crystal above does not move.
What the picture leaves out. Atoms here move on a smooth curve around the step. Real steps have a core with relaxed atom positions, they need thermal energy or stress to move, and several kinds can run along the same boundary at once. Boundaries that are not twins also move by disconnections, usually with larger and more varied (b, h) pairs.
Where this model breaks down
Everything here is geometry: rigid lattices with atoms at their ideal sites. Real boundaries relax their atoms, facet onto low-energy planes, and sit in elastically anisotropic crystals, and none of that is in these pictures. The separate-dislocation picture holds only while the cores stay apart, which is why the sliders stop at small tilts, small misfits and Brandon's limit. Coincidence alone is a weak guide to energy: studies across all five boundary parameters find many low-energy boundaries far from any low Σ, and Mode 5 shows one Σ spanning a twentyfold range. Use CSL and O-lattice theory to work out the dislocations at a boundary you already have, not to predict which boundary will form.
References
Show the 13 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.)
- Olmsted, D. L., Foiles, S. M. & Holm, E. A. "Survey of computed grain boundary properties in face-centered cubic metals: I. Grain boundary energy." Acta Materialia 57, 3694–3703 (2009).
- Tschopp, M. A. & Coleman, S. P. "Symmetric and asymmetric tilt grain boundary structure and energy in Cu and Al (using a novel EAM potential fit with genetic algorithms)." Integrating Materials and Manufacturing Innovation 4, 176–189 (2015).
- Tschopp, M. A. & McDowell, D. L. "Structures and energies of Σ3 asymmetric tilt grain boundaries in copper and aluminium." Philosophical Magazine 87, 3147–3173 (2007).
- Hirth, J. P. & Pond, R. C. "Steps, dislocations and disconnections as interface defects relating to structure and phase transformations." Acta Materialia 44, 4749–4763 (1996).
- Rohrer, G. S. "Grain boundary energy anisotropy: a review." Journal of Materials Science 46, 5881–5895 (2011).
- Watanabe, T. "An approach to grain boundary design for strong and ductile polycrystals." Res Mechanica 11, 47–84 (1984).
- Brandon, D. G. "The structure of high-angle grain boundaries." Acta Metallurgica 14, 1479–1484 (1966).
- Cahn, J. W., Mishin, Y. & Suzuki, A. "Coupling grain boundary motion to shear deformation." Acta Materialia 54, 4953–4975 (2006).
- Han, J., Thomas, S. L. & Srolovitz, D. J. "Grain-boundary kinetics: a unified approach." Progress in Materials Science 98, 386–476 (2018).
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}
}