Geometry / Coordinate System

Geometry of Experiment

Within the PyXMRPackage following definitions for angles and stacking order are used:

../_images/geometry.png

The polarization vectors for \sigma-polarization and \pi-polarization are

\vec{e}_\sigma = \left(\begin{array}{c} 1\\ 0\\ 0\\ \end{array}\right) \quad \, \qquad \vec{e}_\pi = \left(\begin{array}{c} 0\\ \sin\theta\\ \cos\theta\\ \end{array}\right)

The y-z plane is defined / is identical with the scattering plane.

The polarization vectors for left and right circular polarizations are

\vec{e}_{\mathrm{left}} = \frac{1}{2} \left(\begin{array}{c} 1\\ \mathrm{i} \sin \theta\\ \mathrm{i}\cos\theta\\ \end{array}\right) \quad \, \qquad \vec{e}_\mathrm{right} = \frac{1}{2} \left(\begin{array}{c} 1\\ -\mathrm{i}\sin\theta\\ - \mathrm{i}\cos\theta\\ \end{array}\right)

Magnetization

Some parts of PyXMRTool deal with magnetization inside of the sample (SampleRepresentation.MagneticFormfactor and derived classes and SampleRepresentation.MagneticLayerObject). There, the direction of the magnetization is given by the angles \theta_M and \phi_M. The direction of the magnetization as vector \vec{b} is connected to them in the following way:

\vec{b}=\left(\begin{array}{c} \sin\theta_M sin\phi_M\\ \sin\theta_M \cos\phi_M\\ \cos\phi_M\\ \end{array}\right)

So \theta_M is the angle between \vec{b} and the z-direction. \phi_M is the rotation from the scattering plane.

See Macke and Goering 2014, J.Phys.: Condens. Matter 26, 363201. Eq. 11-14 for details.