The rotation between two grains

Take two neighbouring grains, each with its orientation from indexing. The misorientation is the rotation that carries one crystal's axes onto the other's: one axis, one angle, and that pair of numbers is the boundary's identity card. Click any two grains below and read it.

There is a subtlety that catches everyone once: because of crystal symmetry, the same physical boundary can be described by many equivalent axis–angle pairs: 24 of them for cubic. The convention is to quote the disorientation: the description with the smallest angle. For cubic crystals that smallest angle can never exceed 62.8°, which is why every EBSD misorientation histogram you have ever seen ends abruptly at 62.8°; it is not physics, it is bookkeeping.

Boundary interrogator Click one grain, then a second

Grain A Grain B

Pair

–

Disorientation angle

–

degrees · max 62.8° for cubic

Axis (crystal frame)

–

– from that pole

Boundary type

–

– Pick two grains and the boundary tells you what it is.

Is this map "random"? The Mackenzie check

For crystals with no history (no texture, no twinning, orientations drawn from a hat), the distribution of disorientation angles has a known shape, worked out by Mackenzie in 1958. Comparing your map's neighbour-pair histogram against it is the fastest microstructure diagnosis there is:

This map's boundaries (≥ 5°) Mackenzie: random orientations Your selected pair

The spike at 60° is not noise and not texture: it is the twin lamellae, every one contributing a boundary at exactly 60° about ⟨111⟩. A histogram that leaves the Mackenzie curve is a microstructure telling you its history.

Δg = g_B · g_A⁻¹ → θ = min over symmetry of cos⁻¹[(tr(Δg·S) − 1)/2] g_A, g_B are the two orientations; S runs over the 24 cubic rotations. The axis comes from the same reduced rotation. Note what Δg is not: it says nothing about the boundary plane; two boundaries with identical Δg can have different planes and wildly different properties. Five degrees of freedom, and EBSD's standard product measures three.

A trap worth naming

At small angles the angle is measured far better than the axis. The uncertainty in the misorientation axis blows up roughly as 1/θ: for a 1° boundary measured with 0.5° orientation noise, the quoted axis is close to meaningless, and low-angle "tilt vs twist" claims from standard EBSD should be treated accordingly.

The threshold that defines a grain

Software does not detect grains. It detects boundaries, wherever two neighbouring pixels are misoriented by more than a threshold, and calls whatever the boundaries enclose a grain. The threshold is a choice. Drag it and watch the "grain size" of the identical dataset change.

One dataset, any grain size you like The map and the count follow the slider

Low-angle (< 15°) High-angle (≥ 15°)

Grain boundary threshold

Grains detected

–

Mean grain size

–

µm, equivalent circle diameter

– –

The number your paper reports, as a function of the choice you made

Grain count Mean grain size (µm) Current threshold

This dataset has subgrains at 1–4° inside every grain, as any deformed or recovered metal does. Between a 2° and an 8° threshold the grain count changes by several times. Both numbers are defensible. Neither means anything without the threshold printed next to it.

Go deeper: conventions, noise floors, and what to actually report

The de-facto defaults are 5° or 10° for "grain" and 15° for the low-angle/high-angle divide, with 2° as a common subgrain threshold. None of these is physics; 15° is a hand-me-down from the Read–Shockley regime where dislocation cores start to overlap, and 5° vs 10° is habit by instrument vendor. Below about 1–2° you are also fighting the ~0.5° orientation noise from part 1: apparent sub-degree "boundaries" appear everywhere, made of noise. That is why the slider here stops getting meaningful, and real software stops being trustworthy, near its bottom end.

The honest report is the one ASTM E2627 effectively mandates: state the threshold, state the minimum grain-pixel count, state whether twins were merged (module 3), and if the conclusion depends on grain size, show it is robust across reasonable thresholds. A grain size quoted without its threshold is a temperature quoted without units.

Special boundaries and Σ3 twins

Some misorientations are geometrically special: the two lattices, continued through the boundary, share a fraction of their sites. One boundary in Σ3 sites shared is the coincidence-site lattice notation; Σ3, a 60° rotation about ⟨111⟩, is the annealing twin that fills every fcc metal you have ever polished. Special geometry can mean special properties: coherent Σ3{111} twins, with their very low boundary energy, often resist corrosion, cracking and segregation that ordinary boundaries do not. The Σ3 misorientation alone does not guarantee it; the boundary plane matters too.

Σ hunting Brandon's criterion decides what counts

Σ3 (60° ⟨111⟩) Σ9 (38.9° ⟨110⟩) Σ5 (36.9° ⟨100⟩) Ordinary boundary

Brandon tolerance scale

Merge Σ3 twins into their parents

Σ3 length fraction

–

Grains at 5° threshold

–

mean – µm

– –

Tick the merge box and watch the grain count fall and the mean size jump: whether a twin is "a grain" changes every grain-size number downstream. Materials with heavy twinning (brasses, austenitic steels, superalloys) can double their apparent grain count on this one checkbox.

Brandon criterion: Δθ_max = 15° / √Σ a boundary counts as Σn if its disorientation lies within Δθ_max of the exact CSL rotation: 8.7° for Σ3, 6.7° for Σ5, 5° for Σ9. The 15° anchor is the same low-angle limit as before, and the √Σ is a dislocation-spacing argument. The slider above scales this tolerance: tighten it and "special" boundaries evaporate, relax it and half the map goes special. CSL fractions from different papers are only comparable if they used the same criterion.

Go deeper: what Σ does and does not promise

The CSL number is a statement about the two lattices, not about the boundary between them: the actual atomic structure depends on the boundary plane, the missing two degrees of freedom. A coherent Σ3 twin (boundary on the {111} plane) is nearly perfect, with energies an order of magnitude below a general boundary; an incoherent Σ3, same Δg, different plane, behaves much more like an ordinary boundary. That distinction is why grain-boundary engineering works when it does (driving up the coherent twin fraction in fcc alloys genuinely improves corrosion and cracking resistance) and why bare "CSL fraction" statistics oversell it.

Twin-limited microstructures also break naive grain statistics: a heavily twinned austenite has a "grain size" for etchants and optics (twins invisible) and another for EBSD (twins counted), and the two legitimately differ by a factor of two or more. The merge checkbox above is that entire literature dispute in one bit.

The thing to remember

A boundary is not a line on a map: it is a rotation (measured), a plane (usually not), and a threshold (chosen). Every boundary statistic you publish carries all three, whether the caption admits it or not.

Next · Part 4

Texture: pole figures & ODFs, demystified

Rotate a crystal and watch its pole figure move; build a rolling texture grain by grain; see why one pole figure is never enough.

Previous · Part 2

Reading an IPF map without fooling yourself

What the colours encode, the reference-direction trap, and the rotation the colouring cannot see, the one this page's boundaries are made of.

Sources & further reading

J. K. Mackenzie, “Second paper on statistics associated with the random disorientation of cubes,” Biometrika 45, 229 (1958): the random-disorientation distribution and its 62.8° ceiling.
D. G. Brandon, “The structure of high-angle grain boundaries,” Acta Metall. 14, 1479 (1966): the 15°/√Σ criterion used above.
F. J. Humphreys, “Grain and subgrain characterisation by electron backscatter diffraction,” J. Mater. Sci. 36, 3833 (2001): thresholds, noise floors, and honest grain-size practice.
ASTM E2627, Standard Practice for Determining Average Grain Size Using Electron Backscatter Diffraction in Fully Recrystallized Polycrystalline Materials: what to report, and why the threshold goes in the caption.
V. Randle, “Twinning-related grain boundary engineering,” Acta Mater. 52, 4067 (2004): Σ3, coherence, and what CSL fractions do and do not buy.
P. Lin, G. Palumbo, U. Erb, K. T. Aust, “Influence of grain boundary character distribution on sensitization and intergranular corrosion of alloy 600,” Scr. Metall. Mater. 33, 1387 (1995): the classic special-boundaries-resist-attack result.
A. Morawiec, Orientations and Rotations, Springer (2004): the careful mathematics of misorientation, disorientation and fundamental zones.