An orientation says how one crystal sits in the specimen. A misorientation says how two crystals sit relative to each other - it is the rotation that carries the lattice of the first onto the lattice of the second. The specimen has dropped out of the question entirely: turn the sample on the stage and every orientation changes, but no misorientation does.
That independence is why misorientations carry so much of the physics. Whether two grains meet across a twin boundary, whether a lath of martensite grew from a particular parent grain, how much a grain has bent internally - none of these depend on how the sample happened to be mounted, so all of them are naturally expressed as misorientations.
The picture below is a magnesium map whose grain borders are coloured by the misorientation angle across them. Two populations stand out at once: a scatter of small angles, and a set of borders sharing one particular large angle.
plottingConvention.default('y↑→x');
mtexdata twins silent
grains = calcGrains(ebsd('indexed'),'threshold',5*degree);
grains = smoothBoundary(grains,5);
plot(ebsd('indexed'),ebsd('indexed').orientations,'micronbar','off')
hold on
plot(grains.boundary,grains.boundary.misorientation.angle./degree,'linewidth',3)
mtexColorbar('title','misorientation angle in degree')
hold off
The sharp population near 86 degrees is not a coincidence: it is the signature of a twin, one specific misorientation repeated at many independent borders. Finding such populations is most of what this chapter is for.
Two words that are easy to confuse
A crystal has symmetry, so a misorientation is never a single rotation. For a cubic crystal there are 24 ways of labelling the lattice axes of the first crystal and 24 of the second, which gives up to 576 rotations that all describe the same physical relationship. They are called symmetrically equivalent.
Misorientation means any one of them. Disorientation means the particular one with the smallest rotation angle, chosen so that a single number can be quoted. When a boundary is described as "a 60 degree boundary", that 60 degrees is a disorientation angle.
This is also why a misorientation angle can never be compared across phases without care: the set of equivalent rotations depends on the symmetry of both crystals involved.
One further ambiguity is easy to miss. A misorientation has a direction - from the first crystal to the second - and reversing it gives the inverse rotation. Between two grains of the same phase there is nothing that makes either of them first, so the physically meaningful object is the rotation and its inverse taken together. This is called grain exchange symmetry, and it has to be included when misorientations are counted into a distribution, or the same boundary is recorded as two different relationships.
Where to start
Theory is the page to read first. It builds the misorientation from two orientations, shows what symmetry does to it, and introduces the fundamental region - the piece of rotation space in which each physical relationship appears exactly once.
Once misorientations exist as data, the question becomes how they are distributed. The misorientation distribution function, or MDF, plays the same role for misorientations that the ODF plays for orientations, and the MDF page covers it. Its two projections are often more readable than the full function, and each has a page: the angle distribution answers "how far apart are neighbouring crystals", and the axis distribution answers "about which crystal direction do they turn". Comparing either against the distribution expected from randomly oriented crystals is the standard way to show that a population is real.
Next
Misorientations between neighbouring grains are the subject of Grain Boundaries, which is where the CSL and twinning analyses live. Misorientations between a parent phase and its products drive Phase Transitions. The orientations these are all built from are described in Orientations.