# Introduction

EBSD is a powerful tool to determine the sample orientation. During measurement, EBSD collects the user defined data from a specific point of the specimen.These data are stored in the EBSD software as a Raw data. The raw data contains a set of x and y coordinates in which,
* X → Coordinates
* Y → User defined data (eg.Orientation-Euler angles)

The software gives as solution the identified parameters or a zero solution. Zero solutions come from measurements, where the software is unable to detect an EBSP due to several reasons, like sample surface deformations or measurement at grain boundary with overlapping EBSP (Electron backscatter diffraction pattern). The typical raw data is displayed in the following figure.


**BC** refers to **Band contrast** (BC) which is Electron backscattering pattern (EBSP) quality factor derived from the Hough transform. This basically describes the average intensity of the Kikuchi bands with respect to the overall intensity within the EBSP. The value ranges from 0 to 255 which describes low and high contrast.

**BS** refers to the **Band slope** which describes the maximum intensity gradient at the margins of the kikuchi bands.

**MAD** refers to the **Mean angular deviation** which gives the averaged angular misfit in degrees between detected and simulated Kikuchi bands

The Euler angles are the three angles usually φ, ψ, and θ, through which a crystal must be rotated in order to relate the crystallographic axes of the crystal lattice with the sample reference axis. There are three conventions used in Euler angles. They are Bunge, Roe and Kocks.

| Convention | Step 1 | Step 2 | Step 3 | Notations |
| ---------- | ------------------------------------------ | ----------------------------------------------------------------- | -------------------------------- | --------- |
| Bunge | Rotating about ND | Rotating out of plane about [100] | Rotation about [001] | Φ1, Φ, Φ2 |
| Roe | Rotating the crystal about the ND or [001] | Rotating the crystal out of the plane (about [010], or the y-axis | Rotating the crystal about [001] | Ψ, Θ, Φ |
| Kocks|Rotating the crystal about the ND or [001] or z axis|rotating the crystal out of the plane (about [010], or the y axis)|Rotating the crystal anti-clockwise about [001]|Ψ, Θ, ϕ|

The bunge and roe conventions can be related by,

$$
\mathbf{Ψ}_\mathrm{roe}=\mathbf{Ψ}_\mathrm{1,Bunge} -π/2
$$

$$
\mathbf{θ}_\mathrm{Roe}=\mathbf{φ}_\mathrm{Bunge}
$$

$$
\mathbf{φ}_\mathrm{Roe}=\mathbf{Ψ}_\mathrm{2,Bunge}
+π/2
$$

The sample reference frame can be considered as Cs, and the crystal reference plane is Cc. If we want to convert the reference plane from sample to the crystal, the crystal needs to be rotated, that is given by rotation (transformation) matrix G,

$$
\mathbf{C}_\mathrm{c}=\mathbf{G}\cdot\mathbf{C}_\mathrm{s}
$$

Where $\mathbf{G}$ is given by the transformation matrix,

\[
\begin{bmatrix}
cosφ_\mathrm{1}& -sinφ_\mathrm{1}&0\\sinφ_\mathrm{1}&cosφ_\mathrm{1}&0\\0&0&1
\end{bmatrix}

\begin{bmatrix}
1& 0&0\\0& cosΦ&sinΦ\\0&-sinΦ&cosΦ
\end{bmatrix}

\begin{bmatrix}
cosφ_\mathrm{2}& -sinφ_\mathrm{2}&0\\sinφ_\mathrm{2}&cosφ_\mathrm{2}&0\\0&0&1
\end{bmatrix}
\]

\begin{bmatrix}
cosφ_\mathrm{1}cosφ_\mathrm{2}-sinφ_\mathrm{1}sinφ_\mathrm{2}cosΦ
&sinφ_\mathrm{1}cosφ_\mathrm{2}+cosφ_\mathrm{1}sinφ_\mathrm{2}cosΦ
&sinφ_\mathrm{2}\ sinΦ \\
-cosφ_\mathrm{1}sinφ_\mathrm{2}-sinφ_\mathrm{1}cosφ_\mathrm{2}cosΦ
&-sinφ_\mathrm{1}sinφ_\mathrm{2}+cosφ_\mathrm{1}cosφ_\mathrm{2}cosΦ
&cosφ_\mathrm{2}\ sinΦ\\
sinφ_\mathrm{1}sinΦ&-cosφ_\mathrm{1}\ sinΦ&cosΦ
\end{bmatrix}

In crystallography, to describe an orientation, the convention is to take the crystallographic plane normal that is parallel to the specimen normal (e.g. the ND) and a crystallographic direction that is parallel to the long direction (e.g. the RD).


Using the crytsallograophic planes and directions, we can construct a transformation matrix,a - which describes the axis transformation from sample axes to crystal axes and the relation is given in the following table,

| Unit vector | Formula |
| ----------- |:------------------------------------:|
| Plane,n |
=[](https://i.imgur.com/4VlGM41.png) |
| Direction,n |  |
| nˆ × bˆ |  |

The above two transformation matrix can be correlated by the following relation,

$$
\mathbf{a}_\mathbf{ij}=Crystal
$$

\begin{bmatrix}
&sample\\
b_\mathrm{1}&t_\mathrm{1}&n_\mathrm{1}\\
b_\mathrm{2}&t_\mathrm{2}&n_\mathrm{2}\\
b_\mathrm{3}&t_\mathrm{3}&n_\mathrm{3}
\end{bmatrix}

\begin{bmatrix}
cosφ_\mathrm{1}cosφ_\mathrm{2}-sinφ_\mathrm{1}sinφ_\mathrm{2}cosΦ
&sinφ_\mathrm{1}cosφ_\mathrm{2}+cosφ_\mathrm{1}sinφ_\mathrm{2}cosΦ
&sinφ_\mathrm{2}\ sinΦ \\
-cosφ_\mathrm{1}sinφ_\mathrm{2}-sinφ_\mathrm{1}cosφ_\mathrm{2}cosΦ
&-sinφ_\mathrm{1}sinφ_\mathrm{2}+cosφ_\mathrm{1}cosφ_\mathrm{2}cosΦ
&cosφ_\mathrm{2}\ sinΦ\\
sinφ_\mathrm{1}sinΦ&-cosφ_\mathrm{1}\ sinΦ&cosΦ
\end{bmatrix}

1. From this we can extract the miller indices, **(h k l) of the plane and the direction [u v w]**

$$
\mathbf{h}=\mathbf{n}\ \mathbf{sinΦ}\ \mathbf{sinφ}_\mathrm{2}
$$

$$
\mathbf{k}=\mathbf{n}\ \mathbf{sinΦ}\ \mathbf{cosφ}_\mathrm{2}
$$

$$
\mathbf{l}=\mathbf{n}\ \mathbf{cosΦ}\
$$

$$
\mathbf{u}=\mathbf{n}^\mathbf{,}\ \mathbf{cosφ}_\mathrm{1}\ \mathbf{cosφ}_\mathrm{2}-\mathbf{sinφ}_\mathrm{1}\ \mathbf{sinφ}_\mathrm{2}\mathrm(cosΦ)
$$

$$
\mathbf{v}=\mathbf{n}^\mathbf{,}\ \mathbf{-cosφ}_\mathrm{1}\ \mathbf{sinφ}_\mathrm{2}-\mathbf{sinφ}_\mathrm{1}\ \mathbf{cosφ}_\mathrm{2}\mathrm(cosΦ)
$$

$$
\mathbf{w}=\mathbf{n}^\mathbf{,}\ \mathbf{sinΦ}\ \mathbf{sinφ}_\mathrm{1}
$$

2. We can also convert **(hkl) to Euler angles using the following relation**,

$$
Φ=\mathbf{cos}^\mathrm{-1}[\frac{l}{\sqrt{h^2+k^2+l^2}}]
$$

$$
\mathbf{φ}_\mathrm{2}=\mathbf{cos}^\mathrm{-1}[\frac{k}{\sqrt{h^2+k^2}}]
$$

$$
\mathbf{φ}_\mathrm{1}=\mathbf{sin}^\mathrm{-1}[\frac{w}{\sqrt{u^2+v^2+w^2}}] \ [\frac{\sqrt{h^2+k^2+l2}}{\sqrt{h^2+k^2}}]
$$

**Reference:**

Texture analysis in materials science: mathematical methods
HJ Bunge - 2013