Misorientation

misorientation allows to determine orientation relationship and misorientation between two crystals It allows:

  • Display two crystals at the same time on a stereographic projection which allows the easy determination of orientation relationship

  • Orient two crystals independently

  • Determine misorientation between two grains (same crystals)

Misorientation

When two crystals are separated by a grain boundary, the neighboring crystals may exhibit a specific orientation relationship that can be defined by an interface and a couple rotation axis, i.e. a common crystal direction, misorientation angle.

Considering the orientation of grains A and B defined by their Euler angles. Their orientation in the reference frame is defined by the orientation matrix O_1 and O_2. The misorientation matrix is given by:

M=O_1 O_2^{-1}

As M is a rotation matrix, one could extract the couple angle-axis, (\theta,\vec{u}) according to:

\theta=\arccos \left(\frac{tr(M)-1}{2}\right)

and

M\vec{u}=\vec{u}

However, because of crystal symmetry, the misorientation between two grains is not unique, i.e. equivalent (\vec{u},\theta) exist. The misorientation axis should then be modified by:

M=O_1O_SO_2^{-1}

where O_S are the symmetry operations of the crystal. For cubic, it consists of:

  • Rotations along \langle 100 \rangle of 90°, 180° and 270° (9 rotations)

  • Rotations along \langle 110 \rangle of 180° (6 rotations)

  • Rotations along \langle 111 \rangle of 120° and 240° (8 rotations)

  • Identity operation

Thus 24 symmetry operations can be applied, leading to 24 (\vec{u},\theta) couples.

Equivalent considerations can be made for other crystal systems.

Interface

The interface is composed of 3 tabs.

  • the settings tab is used to enter the 2 crystal inputs, the microscopic settings regarding rotation between tilt axis and image, and the layout. Refer to stereoproj for the details information.

  • the stereo-proj tab is the plotting area

  • misorientation tab display the misorientation between crystal (if identical crystal are selected)

Procedure

Plotting

  • Set the crystal parameter and layout in the setting tab. Refer to stereoproj for the details.

  • Enter the crystal 1 and 2 orientation in the Euler Angles fields

  • Plot the stereographic projections using the Plot button

Overlay of an aluminium and an zirconium crystal.

  • Same additional features as in stereoproj are available: adding poles/directions, planes... and can be performed independently on crystal 1 or 2 by using the Switch button

  • Rotations can be performed along x,y,z or along a pole/direction (g) entered in the Pole/direction field, either in crystal 1 or 2

Misorientation

For 2 crystals of the same structure, the misorientation can be computed after plotting the stereographic projection.

In the misorientation tab:

  • press the Find misorientation Plot Angle/Axis button

  • the rotation axes in both crystal are displayed along with the rotation angle

  • if the uvw button is ticked in the setting tab, the axes will be given as directions and not plane normals.

  • in the stereographic tab, the axes appear as black dots.

  • ticking the show numbers or show axes will display the number corresponding to the angle/axis couples and the axis indices in crystal 1 on the stereographic projection, respectively.