# Bicrystal Definition¶

## Crystallographic properties of a bicrystal¶

A bicrystal is formed by two adjacent crystals separated by a grain boundary.

Five macroscopic degrees of freedom are required to characterize a grain boundary [3], [5], [6] and [7] :
• 3 for the rotation between the two crystals;
• 2 for the orientation of the grain boundary plane defined by its normal $$n$$.

The rotation between the two crystals is defined by the rotation angle $$\omega$$ and the rotation axis common to both crystals $$[uvw]$$.

Using orientation matrix of both crystals obtained by EBSD measurements, the misorientation or disorientation matrix $$(\Delta g)$$ or $$(\Delta g_\text{d})$$ is calculated [4] and [2] :

(1)$\Delta g = g_\text{B}g_\text{A}^{-1} = g_\text{A}g_\text{B}^{-1}$
(2)$\Delta g_\text{d} = (g_\text{B}*CS)(CS^{-1}*g_\text{A}^{-1}) = (g_\text{A}*CS)(CS^{-1}*g_\text{B}^{-1})$

Disorientation describes the misorientation with the smallest possible rotation angle and $$CS$$ denotes one of the symmetry operators for the material [1].

The Matlab function used to set the symmetry operators is : sym_operators.m

The orientation matrix $$g$$ of a crystal is calculated from the Euler angles ($$\phi_{1}$$, $$\Phi$$, $$\phi_{2}$$) using the following equation :

(3)$\begin{split}g = \begin{pmatrix} \cos(\phi_{1})\cos(\phi_{2})-\sin(\phi_{1})\sin(\phi_{2})\cos(\Phi) & \sin(\phi_{1})\cos(\phi_{2})+\cos(\phi_{1})\sin(\phi_{2})\cos(\Phi) & \sin(\phi_{2})\sin(\Phi) \\ -\cos(\phi_{1})\sin(\phi_{2})-\sin(\phi_{1})\cos(\phi_{2})\cos(\Phi) & -\sin(\phi_{1})\sin(\phi_{2})+\cos(\phi_{1})\cos(\phi_{2})\cos(\Phi) & \cos(\phi_{2})\sin(\Phi) \\ \sin(\phi_{1})\sin(\Phi) & -\cos(\phi_{1})\sin(\Phi) & \cos(\Phi) \\ \end{pmatrix}\end{split}$

The orientation of a crystal (Euler angles) can be determined via electron backscatter diffraction (EBSD) measurement or via transmission electron microscopy (TEM).

The Matlab function used to generate random Euler angles is : randBunges.m

The Matlab function used to calculate the orientation matrix from Euler angles is : eulers2g.m

The Matlab function used to calculate Euler angles from the orientation matrix is : g2eulers.m

Then, from this misorientation matrix ($$\Delta g$$), the rotation angle ($$\omega$$) and the rotation axis $$[u, v, w]$$ can be obtained by the following equations :

(4)$\omega = \cos^{-1}((tr(\Delta g)-1)/2)$
(5)$\begin{split}u = \Delta g_{23} - \Delta g_{32} \\ v = \Delta g_{31} - \Delta g_{13} \\ w = \Delta g_{12} - \Delta g_{21}\end{split}$

The Matlab function used to calculate the misorientation angle is : misorientation.m

The grain boundary plane normal $$n$$ can be determined knowing the grain boundary trace angle $$\alpha$$ and the grain boundary inclination $$\beta$$.

The grain boundary trace angle is obtained through the EBSD measurements (grain boundary endpoints coordinates) and the grain boundary inclination can be assessed by a serial polishing (chemical-mechanical polishing or FIB sectioning), either parallel or perpendicular to the surface of the sample (see Figure 5).

Figure 4 Schematic of a bicrystal.

Figure 5 Screenshot of the Matlab GUI used to calculate grain boundary inclination.