Misorientation describe the relative orientation of two crystal with respect to each other. Those crystal may be of the same phase or of different phases. Misorientation are used to describe orientation relationships at grain boundaries, orientation relationships during phase transformations or twinning and orientation gradients within grains.

## Misorientations within the same phase

Misorientation describes the relative orientation of two grains with respect to each other. Important concepts are twinning and CSL (coincidence site lattice) misorientations. To illustrate this concept at a practical example let us first import some Magnesium EBSD data.

Next we plot the grains together with their mean orientation and highlight grain 74 and grain 85

After extracting the mean orientation of grain 74 and 85

we may compute the misorientation angle between both orientations by

Note that the misorientation angle is computed by default modulo crystal symmetry, i.e., the angle is always the smallest angles between all possible pairs of symmetrically equivalent orientations. In our example this means that symmetrisation of one orientation has no impact on the angle

The misorientation angle neglecting crystal symmetry can be computed by

We see that the smallest angle indeed coincides with the angle computed before.

## Misorientations

Remember that both orientations `ori1`

and `ori2`

map crystal coordinates onto specimen coordinates. Hence, the product of an inverse orientation with another orientation transfers crystal coordinates from one crystal reference frame into crystal coordinates with respect to another crystal reference frame. This transformation is called misorientation

In the present case the misorientation describes the coordinate transform from the reference frame of grain 85 into the reference frame of crystal 74. Take as an example the plane \(\{11\bar{2}0\}\) with respect to the grain 85. Then the plane in grain 74 which aligns parallel to this plane can be computed by

Conversely, the inverse of `mori`

is the coordinate transform from crystal 74 to grain 85.

## Coincident lattice planes

The coincidence between major lattice planes may suggest that the misorientation is a twinning misorientation. Lets analyze whether there are some more alignments between major lattice planes.

we observe an almost perfect match for the lattice planes \(\{11\bar{2}0\}\) to \(\{\bar{2}110\}\) and \(\{1\bar{1}01\}\) to \(\{\bar1101\}\) and good coincidences for the lattice plane \(\{1\bar100\}\) to \(\{0001\}\) and \(\{0001\}\) to \(\{0\bar661\}\). Lets compute the angles explicitly

## Twinning misorientations

Lets define a misorientation that makes a perfect fit between the \(\{11\bar20\}\) lattice planes and between the \(\{10\bar11\}\) lattice planes

and plot the same figure as before with the exact twinning misorientation.

## Highlight twinning boundaries

It turns out that in the previous EBSD map many grain boundaries have a misorientation close to the twinning misorientation we just defined. Lets Lets highlight those twinning boundaries

From this picture we see that large fraction of grain boundaries are twinning boundaries. To make this observation more evident we may plot the boundary misorientation angle distribution function. This is simply the angle distribution of all boundary misorientations and can be displayed with

From this we observe that we have about 50 percent twinning boundaries. Analogously we may also plot the axis distribution

which emphasizes a strong portion of rotations about the \((\bar12\bar10)\) axis.

## Phase transitions

Misorientations may not only be defined between crystal frames of the same phase. Lets consider the phases Magnetite and Hematite.

The phase transition from Magnetite to Hematite is described in literature by \(\{111\}_m\) parallel \(\{0001\}_h\) and \{\bar101}_m\( parallel \)\{10\bar10\}_h$. The corresponding misorientation is defined in MTEX by

Assume a Magnetite grain with orientation

Then we can compute all variants of the phase transition by

and the corresponding pole figures by