Reference frames

Sample-detector geometry

Fig. 1 and Fig. 2 show EBSD reference frames used in kikuchipy, all of which are right handed. They are based on definitions presented in [Britton2016] and the accompanying supplementary material.

_images/detector_sample_geometry.png

Fig. 1 (a) An EBSD experiment showing the orientation of the crystal reference frame, \(x_{euler}-y_{euler}-z_{euler}\), attached to the sample. The RD-TD-ND crystal reference frame used by EDAX TSL is shown for reference. An EBSD pattern on the detector screen is viewed from behind the screen towards the sample. (b) How the EBSD map appears within the data collection software, with the crystal reference frame and the scanning reference frame, \(x_{scan}-y_{scan}-z_{scan}\), attached. (c) The relationship between the crystal reference frame and the detector reference frame, \(x_{detector}-y_{detector}-z_{detector}\), with the projection center highlighted. The detector tilt \(\theta\) and sample tilt \(\sigma\), in this case \(10^{\circ}\) and \(70^{\circ}\), respectively, are also shown.

_images/detector_gnomonic_coordinates.jpg

Fig. 2 The EBSD pattern in Fig. 1 (a) as viewed from behind the screen towards the sample (left), with the detector reference frame the same as in (c) with its origin (0, 0) in the upper left pixel. The detector pixels’ gnomonic coordinates can be described with a calibrated projection center (PC) (right), with the gnomonic reference frame origin (0, 0) in (PCx, PCy). The circles indicate the angular distance from the PC in steps of \(10^{\circ}\).

The EBSD detector

All relevant parameters for the sample-detector geometry are stored in an EBSDDetector object:

>>> from kikuchipy.detectors import EBSDDetector
>>> det = EBSDDetector(
...     shape=(60, 60),
...     pc=[0.421, 0.779, 0.505],
...     convention="tsl",
...     pixel_size=70,  # microns
...     binning=8,
...     tilt=0,
...     sample_tilt=70,
... )
>>> det
EBSDDetector (60, 60), px_size 70 um, binning 8, tilt 0, pc
(0.421, 0.221, 0.505)
>>> det.pc_tsl()
array([[0.421, 0.779, 0.505]])

The projection/pattern center (PC) is stored internally in the Bruker convention:

  • PCx is measured from the left border of the detector in fractions of detector width.

  • PCy is measured from the top border of the detector in fractions of detector height.

  • PCz is the distance from the detector scintillator to the sample divided by pattern height.

Above, the PC was passed in the EDAX TSL convention. Passing the PC in the Bruker, Oxford, or EMsoft v4 or v5 convention is also supported. Likewise, the PC can be returned in all conventions via pc_emsoft() and similar. Conversions between conventions are implemented as described in [JPDeGraef19]. The unbinned pixel size \(\delta\), binning factor \(b\) and number of pixel rows \(s_y\) and columns \(s_x\) are needed to convert a PC between the EMsoft and Bruker conventions:

  • EDAX TSL or Oxford to Bruker

\[[PC_x, PC_y, PC_z] = [x^*, 1 - y^*, z^*].\]
  • EMsoft to Bruker, with \(v = -1\) for EMsoft v5 and \(+1\) for v4

\[[PC_x, PC_y, PC_z] = \left[ \frac{1}{2} + v\frac{x_{pc}}{s_x b}, \frac{1}{2} - \frac{y_{pc}}{s_y b}, \frac{L}{s_y \delta b} \right].\]

The detector can be plotted to show whether the average PC is placed as expected using plot() (see its docstring for a complete explanation of its parameters):

>>> det.plot()

This will produce a figure similar to the left panel in Fig. 2, without the pattern, arrows and colored labels. Passing a numpy array in the pattern parameter places a pattern on the detector.

Multiple PCs with a 1D or 2D navigation shape can be passed to the pc parameter upon initialization, or can be set directly. This gives the detector a navigation shape (not to be confused with the detector shape) and a navigation dimension (maximum of two):

>>> det.pc = np.ones((3, 4, 3)) * [0.421, 0.779, 0.505]
>>> det.navigation_shape
(3, 4)
>>> det.navigation_dimension
2
>>> det.pc = det.pc[0, 0]
>>> det.navigation_shape
(1,)

Note

The offset and scale of HyperSpy’s axes_manager is fixed for a signal, meaning that we cannot let the PC vary with scan position if we want to calibrate the EBSD detector via the axes_manager. The need for a varying PC was the main motivation behind the EBSDDetector class.

The right panel in Fig. 2 shows the detector plotted in the gnomonic projection using plot(). We assign 2D gnomonic coordinates (\(x_g\), \(y_g\)) in a gnomonic projection plane parallel to the detector screen to a 3D point (\(x_d\), \(y_d\), \(z_d\)) in the detector frame as

\[x_g = \frac{x_d}{z_d}, \qquad y_g = \frac{y_d}{z_d}.\]

The detector bounds and pixel scale in this projection, per navigation point, are stored with the detector:

>>> det.bounds  # In pixels
array([ 0, 59,  0, 59])
>>> det.gnomonic_bounds
array([[-0.83366337,  1.14653465, -1.54257426,  0.43762376]])
>>> det.x_range
array([[-0.83366337,  1.14653465]])
>>> det.r_max  # Largest radial distance to PC
array([[1.92199819]])