The angle two grains differ by

Two grains meet at a boundary. How far apart are they?

The question sounds like it has an obvious answer — take the rotation carrying one grain’s axes to the other’s and read off its angle — and that answer is wrong, or rather it is one of several hundred answers with no reason to prefer it.

A crystal’s axes are not labelled #

An orientation is a rotation carrying a reference frame to a crystal’s axes. But if the crystal has a four-fold axis, then turning it by ninety degrees about that axis produces the same crystal, sitting in the same place, indistinguishable by any measurement. The rotation is different and the orientation is not.

So an orientation is not a rotation. It is a coset: a rotation together with everything the crystal’s own rotation group can do to it, and the cubic class has twenty-four of those.

A misorientation between two grains is worse, because either grain’s axes may be relabelled. The relative rotation q and the rotation g₁ q g₂ describe the same physical relationship for any g₁ and g₂ in the group — a double coset — and for the cubic class that is 24 × 24 = 576 rotations describing one misorientation.

The disorientation #

Among the descriptions, one has the smallest rotation angle, and that angle is the disorientation. It is the honest answer, it is what any two people measuring the same boundary will agree on, and it is computed here by taking the minimum over the group twice rather than by choosing a convenient representative.

That distinction matters more than it looks. The literature has conventions — a fundamental zone in orientation space, a canonical axis-angle pair — and those conventions are shortcuts to this minimum. Taking the minimum directly is slower and needs no convention, which means the result can be compared against any convention rather than depending on one.

The immediate consequences are two lines of arithmetic that the computation confirms. The identity has disorientation nought, and so does every operation of the crystal’s own group — a grain is not misoriented with respect to itself, however its axes are labelled. Both are checked, and a minimisation that failed either would be minimising something else.

The largest there is #

The interesting quantity is the maximum: how far apart two grains of a given class can possibly be.

For a crystal with no rotational symmetry it is a hundred and eighty degrees, because the only description is the rotation itself and a half turn is the furthest any rotation goes. Every symmetry the crystal has adds descriptions, the minimum is taken over more of them, and the maximum comes down.

For the cubic class it is 62.8°. Two cubic grains, however they are cut, however they grew, however anybody turns them, differ by at most sixty-three degrees. That is not a statement about any material; it is a fact about the group.

The refinement is worth a note because it is the difference between a plausible number and a usable one. Sampling twenty thousand random orientations gets to about 62.4°, which is a lower bound and looks like an answer. Climbing locally from the best sample takes it to 62.798°, and it stops there — which is the value the literature records. The essay reports it as a refined lower bound rather than as a solved maximum, because the exact value sits at a vertex of the fundamental zone and is an algebraic number this computation does not solve for.

Why the arithmetic is done in quaternions #

Rotations can be written as three-by-three matrices, and the minimisation above could be done with them. It is done with quaternions instead, for three reasons that are worth separating.

The first is that the angle is one line. A unit quaternion has a real part cos(θ/2), so the rotation angle is 2 arccos|w| and the minimisation over five hundred and seventy-six products is a minimisation over five hundred and seventy-six real parts. With matrices it would be a trace, which is the same information and more arithmetic to extract.

The second is that composition is cheap and exact. A product of two unit quaternions is sixteen multiplications, and — importantly here — the crystal’s rotation group is a set of quaternions obtained by closing a couple of generators, which is exactly what the double group of a crystal class is built from. The group used here is that construction taken modulo the sign, since a rotation has two quaternions above it and both describe the same rotation.

The third is the one that ties this to the rest of the collection. Rational rotations of space are integer quaternions — that is the parametrisation the cubic coincidence series is enumerated from, and the same one a companion matrix supplies in a different setting. So the coincidence rotations arrive as quaternions already, and putting them on the disorientation scale needs no conversion at all.

What the number is for #

The disorientation is not an abstraction; it is the coordinate a boundary is filed under, and several essays in this collection use it without naming it.

A small angle is a row of dislocations is the low end of the scale: below about fifteen degrees a boundary is a discrete array of dislocations with good crystal between them, and its spacing is the disorientation through Frank’s formula. Above that the dislocation cores overlap and the description stops meaning anything — so the disorientation is the parameter that decides which of two completely different pictures of a boundary applies.

A beat is not a period is the same scale seen through a microscope: the moiré fringes of a twist boundary have a spacing set by the same angle, and the fringes are the dislocation network. The three descriptions — an angle, a dislocation spacing, a fringe period — are one number in three units.

The index and the angle a twin misses by uses it at the other end: a twin law is an exact misorientation and a real twin sits near it, and the discrepancy is quoted as an angle, which has to be a disorientation to be comparable between materials. The dislocations a boundary allows then works inside a fixed disorientation and asks what can move along the boundary at that angle.

So the same coordinate carries a boundary from “is it special” through “what does it look like” to “what can it do”, and it is the coordinate that has five hundred and seventy-six spellings unless somebody takes the minimum.

Orientation and misorientation are different quotients #

One more distinction, because conflating the two is easy and the numbers differ.

An orientation is a rotation modulo the crystal group on one side, so there are |G| descriptions and the space of orientations is 1/|G| of the space of rotations. A misorientation is modulo the group on both sides, so there are up to |G|² descriptions — and the space of misorientations is smaller again, though not by a factor of |G|², because the double cosets are not all the same size.

That asymmetry has a physical reading. Measuring the orientations of two grains separately and subtracting gives a rotation that has to be reduced twice; measuring the boundary between them gives the relationship directly and still has to be reduced twice, since neither grain’s axes were labelled either. There is no measurement that avoids the reduction, which is why the disorientation is the quantity every convention is trying to name.

It also explains why the maximum for misorientations is so much smaller than a half turn while the maximum for orientations is not interesting at all: an orientation has no natural “distance from nothing” because there is no reference orientation, and the whole question only exists for pairs.

What actually decides the maximum #

Reading down the table, the maximum does not simply fall as the group grows. The six-fold class 6 has six rotations and a maximum of a hundred and eighty degrees; the class 222 has four and a maximum of a hundred and twenty.

What decides it is the axes, not the order. A group whose rotations all share one axis can do nothing about a rotation perpendicular to that axis: turning the crystal about its own axis leaves a perpendicular half turn a perpendicular half turn, so the minimum over the group is still a hundred and eighty degrees. Six-fold symmetry about a single axis buys nothing at all against the worst case.

A group with axes pointing in several directions has somewhere to move any rotation towards, and the more of the sphere the axes cover the further the maximum comes down: three perpendicular two-folds reach a hundred and twenty degrees with four operations, and the cubic class’s thirteen axes reach sixty-three with twenty-four.

What a random pair looks like #

The maximum is the extreme; the ordinary case is a distribution.

A random pair of cubic grains has a disorientation of about forty-one degrees on average, and the distribution is strongly peaked near forty-five. That curve is what a measured texture is compared against: a material whose grain boundaries follow it has no preferred orientation relationship, and a departure from it is the evidence that something — growth, deformation, a transformation — produced one.

The curve is a consequence of the symmetry and of nothing else, which is why it is worth deriving rather than measuring. Any deviation a laboratory sees is a fact about the sample; the shape it deviates from is a fact about the cubic group.

Two features of it are worth naming because they are easy to misread. The curve rises from zero rather than starting high: a disorientation near zero is rare, not because small-angle boundaries are unlikely in a material but because there is very little of orientation space near the identity, which is a volume effect and not a preference. And it stops rather than tailing off: there is no tail past sixty-three degrees at all, so a histogram that has one is reporting a measurement error, a wrong symmetry, or an unreduced angle. A curve with a hard edge is a useful curve, because departures from it at the edge are unambiguous.

Where the special boundaries fall #

The boundaries a metallurgist cares about are the coincidence ones — the misorientations at which the two grains share a sublattice, indexed by Σ, which this collection enumerates from integer quaternions. Putting their disorientation angles on the same scale is a one-line computation once the minimisation exists.

Two things come out of it.

The spread is worth reading carefully, because the two orderings are genuinely independent rather than merely different. Sorting the coincidence rotations by index gives one list; sorting them by disorientation gives another; and the correlation between the two is close to nothing. A boundary is special because its lattices share points, which is an arithmetic property of a rotation, and it is high-angle or low-angle because of how far that rotation turns, which is a metric property — and the two have no reason to agree.

A low index is not a small angle. Σ3 — the coherent twin, the most special boundary there is — sits at sixty degrees, near the top of the range. Σ13 sits at twenty-three. The ordering by index and the ordering by angle have almost nothing to do with each other, so “a high-angle boundary” and “a general boundary” are not the same category however often they are used as though they were.

And the angles the literature quotes for them are already disorientations. Every Σ rotation in the standard table turns out to be listed at its minimum angle rather than at some other of its five hundred and seventy-six descriptions. That is a convention this computation can confirm rather than assume — and confirming a convention is worth doing, because a table that mixed conventions would be impossible to detect by reading it.

What this does not do #

It is the proper classes only. Improper operations — mirrors, inversion — do not act on orientations in the same way, because a mirror maps a right-handed frame to a left-handed one and no rotation does. A crystal whose class contains a mirror still has only its rotations available for relabelling axes, so the eleven proper classes are the whole of what matters here, and that is a fact rather than a restriction of scope.

And the maximum is a refined sample. The refinement stops moving, and it lands on the recorded values to three decimals across six classes, which is strong evidence and is not a proof. Solving for the vertex of the fundamental zone exactly would be a different computation and is not done here.

And the sampling is a stated stream. Every random orientation above comes from one seeded generator by Shoemake’s construction, which is uniform on the rotations rather than uniform in any set of angles — an easy thing to get wrong, since sampling three Euler angles uniformly is not uniform on rotations and would bias every number in the distribution. The seed is fixed so the figures reproduce.

And nothing here is about energy. Which boundaries a material actually contains depends on how it grew and what it costs to make one, and both are outside this collection. What the arithmetic supplies is the range the answer must lie in and the distribution it would follow if nothing preferred anything — the usual division, and the usual reason the null case is worth computing exactly.

A boundary between two different crystals #

Everything above assumes the two grains are the same substance, so that one group serves for both relabellings. When they are not — a film on a substrate, two phases meeting inside an alloy — the double coset is over two groups, g₁ q g₂ with g₁ in one crystal’s group and g₂ in the other’s, and the count of descriptions is the product of the two orders rather than the square of one.

Nothing about the minimisation changes; it is still a minimum over a finite set. What changes is that the maximum is no longer a property of one class. A cubic film on a hexagonal substrate has 24 × 12 = 288 descriptions and a maximum somewhere between the two classes’ own — smaller than the hexagonal one because the cubic group is helping, larger than the cubic one because the hexagonal group has less to offer.

That is the setting epitaxy is a measurement works in, and it is why an epitaxial relationship is quoted as a pair of planes and directions rather than as an angle: with two groups and two lattices there is no single number that reduces to a canonical form, and the honest description names what is parallel to what.

The same applies inside one material at a phase transition. How many domains a transition makes counts the orientation states a descent produces, and the relationships between them are misorientations of exactly this kind — one group above, another below, and the number of distinct relationships the index of one in the other. The disorientation is what makes those relationships comparable, and the index is what makes them countable, and the two questions are asked of the same object.

The one thing to carry #

A misorientation is a double coset and an angle quoted for one is a choice unless the minimum was taken. Five hundred and seventy-six numbers describe the same physical boundary in a cubic material, they span from a few degrees to a hundred and seventy-nine, and only one of them is reproducible.

That is a small piece of bookkeeping with a large consequence for reading anybody’s data. A boundary reported at a hundred and twenty degrees in a cubic material is not a boundary at all in the sense the number suggests — it is a boundary whose disorientation is at most sixty-three, described from an unlucky corner of the symmetry. The maximum is the check: any angle above it, in any cubic measurement, is a description rather than a measurement.