Coordinate System Conventions

Axis Definitions

The Scout-and-Scan Coordinate Convention figure shows how SampleX, SampleY and SampleZ are defined by the Scout-and-Scan Control System when the sample is mounted on the ZEISS Xradia Versa or the ZEISS Xradia CrystalCT and the Sample Theta stage is at 0°. When the Sample Theta stage is at -90°, SampleZ moves orthogonal to the beam line and SampleX moves on the beam line. This coordinate system is also used by the Acquisition Wizard to operate the microscope consistently.

Scout-and-Scan Coordinate Convention

Refer to the ZEISS sample axis definitions outlined in the ”Axis Definitions” in Chapter 1 of the instruments user guide. GrainMapper3D operates with a relative coordinate system attached to the imported absorption volume as pictured in GrainMapper3D Coordinate Convention. By default the origin (0,0,0) of the GrainMapper3D coordinate system is the mean SampleX, SampleY and SampleZ stage position of the reconstructed absorption volume.

GrainMapper3D Coordinate Convention

For a sample coordinate \(({x,y,z})\) given in one or the other coordinate system, the transformations are:

\[\begin{split}\left(\begin{array}{c} x\\ y\\ z \end{array}\right)_{\text{Xnovo}}=\left(\begin{array}{ccc} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{array}\right)\left(\begin{array}{c} x\\ y\\ z \end{array}\right)_{\text{ZEISS}}=\left(\begin{array}{c} z\\ -x\\ -y \end{array}\right)_{\text{ZEISS}}\end{split}\]

and

\[\begin{split}\left(\begin{array}{c} x\\ y\\ z \end{array}\right)_{\text{ZEISS}}=\left(\begin{array}{ccc} 0 & -1 & 0\\ 0 & 0 & -1\\ 1 & 0 & 0 \end{array}\right)\left(\begin{array}{c} x\\ y\\ z \end{array}\right)_{\text{Xnovo}}=\left(\begin{array}{c} -y\\ -z\\ x \end{array}\right)_{\text{Xnovo}}.\end{split}\]

Crystal Reference Frame

The crystal reference frame used is defined according to the international convention,

\[\mathbf{x} \parallel \mathbf{a},\qquad \mathbf{z} \parallel \mathbf{c}^{*},\]

which for unit cell parameters \(({a,b,c,\alpha,\beta,\gamma})\) results in the direct structure matrix

\[\begin{split}\text{B} = \begin{pmatrix} a & {b\cos\gamma} & {c\cos\beta} \\ 0 & {b\sin\gamma} & {c\csc\gamma\left( {\cos\alpha - \cos\beta\cos\gamma} \right)} \\ 0 & 0 & {c\csc\gamma\sqrt{1 - \cos^{2}\alpha - \cos^{2}\beta - \cos^{2}\gamma + 2\cos\alpha\cos\beta\cos\gamma}} \end{pmatrix}\end{split}\]

and the reciprocal structure matrix \(\mathbf{B}^{*} = \mathbf{B}^{- 1}\), respectively. A direct space vector \(\mathbf{u} = (u,v,w)^{T}\) has the coordinate

\[\mathbf{u}' = \mathbf{B}\mathbf{u}\]

in the crystal reference frame. A reciprocal lattice vector \(\mathbf{h} = (h,k,l)^{T}\) has the coordinate

\[\mathbf{h}'^{T} = \mathbf{h}^{T}\mathbf{B}^{*}\]

in the crystal reference frame.

Orientation Convention

The rotation matrix \(\mathbf{U}\) maps a vector \(\mathbf{h}'\) from the crystal reference frame onto a vector \(\mathbf{r}\) in the sample reference frame,

\[\mathbf{U}\mathbf{h}' = \mathbf{r},\]

where the sample reference frame is defined according to the axis definitions in GrainMapper3D Coordinate Convention.

Note

For comparison with common post-processing tools for crystallographic data, the GrainMapper3D orientation convention is the same as used by MTEX, and the inverse of that adapted by DREAM.3D.