Coordinate systems

This page defines the coordinate systems in which easistrain defines the scattering vector.

Sample reference frame

Right-handed Eucledian reference frame \((\hat{S}_1, \hat{S}_2, \hat{S}_3)\) with

  • \(\hat{S}_1\): physically meaningfully direction (rolling direction, machining direction, welding direction, …)

  • \(\hat{S}_3\): sample surface normal

Laboratory reference frame

Right-handed Eucledian reference frame \((\hat{L}_1, \hat{L}_2, \hat{L}_3)\) with

  • \(\hat{L}_1\): beam direction

  • \(\hat{L}_3\): inverse direction of gravity

Note that \(\hat{L}_2\) points to the left when looking downstream and \(\hat{L}_3\) points upwards.

In the code this reference frame is also referred to as the goniometer reference frame.

Scattering vector

The cartesian coordinates \(Q\) of the normalized scattering vector \(\hat{q}\) in the sample reference frame \((\hat{S}_1, \hat{S}_2, \hat{S}_3)\) are given by

\[\begin{split}Q_\text{sample} = \begin{bmatrix} \cos \phi \sin \psi \\ \sin \phi \sin \psi \\ \cos \psi \end{bmatrix}\end{split}\]

  • \(\phi\): azimuth angle of \(\hat{q}\) in spherical coordinates

  • \(\psi\): polar angle of \(\hat{q}\) in spherical coordinates

The cartesian coordinates \(Q\) of the normalized scattering vector \(\hat{q}\) in the laboratory reference frame \((\hat{L}_1, \hat{L}_2, \hat{L}_3)\) are given by

\[\begin{split}Q_\text{lab} = \begin{bmatrix} -\sin \theta \\ -\cos \theta \sin \delta \\ \cos \theta \cos \delta \end{bmatrix}\end{split}\]

  • \(\theta\): half the angle between incident beam (along \(\hat{L}_1\)) and scattered beam

  • \(\delta\): angle of scattered beam in the \(\hat{L}_3\times-\hat{L}_2\) plane (careful: \(\delta=0^\circ\) is \(\hat{L}_3\) and

\(\delta=90^\circ\) is \(-\hat{L}_2\))

The easistrain project refers to a horizontal and vertical detector as

  • horizontal detector (\(\delta=-90^\circ\)): scattering plane is in the synchrotron plane and the detector is positioned on the left when looking downstream

\[\begin{split}Q_\text{lab,H} = \begin{bmatrix} -\sin \theta_H \\ \cos \theta_H \\ 0 \end{bmatrix}\end{split}\]

  • vertical detector (\(\delta=0^\circ\)): scattering plane is perpendicular to the synchrotron plane and the detector is positioned above the synchrotron plane

\[\begin{split}Q_\text{lab,V} = \begin{bmatrix} -\sin \theta \\ 0 \\ \cos \theta \end{bmatrix}\end{split}\]

The activate transformations of \(\hat{q}\) between cartesian coordinates in laboratory to sample frame are given by

\[Q_\text{lab} = R_2(-\omega) \cdot R_1(\chi) \cdot R_3(-\varphi) \cdot Q_\text{sample}\]
\[Q_\text{sample} = R_3(\varphi) \cdot R_1(-\chi) \cdot R_2(\omega) \cdot Q_\text{lab}\]

  • \(\chi\): activate rotation of the sample around \(\hat{L}_1\)

  • \(\omega\): activate rotation of the sample around \(-\hat{L}_2\)

  • \(\varphi\): activate rotation of the sample around \(-\hat{L}_3\)

  • \(R_i\): activate transformation matrix around axis \(i\)

For example \(\chi=0^\circ\), \(\omega=90^\circ\) and \(\varphi=0^\circ\) positions the sample surface perpendicular to the beam (\(\hat{S}_3 = -\hat{L}_1\)).