# Reference frames¶

## Sample-detector geometry¶

The figure below shows the sample reference frame and the detector reference frame used in kikuchipy, all of which are right handed. In short, the sample reference frame is the one used by EDAX TSL, RD-TD-ND, while the pattern center is defined as in the Bruker software.

In (a) (lower left), a schematic of the microscope chamber shows the definition of the sample reference frame, RD-TD-ND. The $$x_{euler}-y_{euler}-z_{euler}$$ crystal reference frame used by Bruker is shown for reference. An EBSD pattern on the detector screen is viewed from behind the screen towards the sample. The inset (b) shows the detector and sample normals viewed from above, and the azimuthal angle $$\omega$$ which is defined as the sample tilt angle round the RD axis. (c) shows how the EBSD map appears within the data collection software, with the sample reference frame and the scanning reference frame, $$x_{scan}-y_{scan}-z_{scan}$$, attached. Note the $$180^{\circ}$$ rotation of the map about ND. (d) shows the relationship between the sample 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.

The above figure shows the EBSD pattern in the sample reference frame figure (a) as viewed from behind the screen towards the sample (left), with the detector reference frame the same as in (d) 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 ($$PC_x, PC_y$$). 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 kikuchipy.detectors.EBSDDetector instance. Let’s first import necessary libraries and a small Nickel EBSD test data set

[1]:

# Exchange inline for notebook or qt5 (from pyqt) for interactive plotting
%matplotlib inline

import matplotlib.pyplot as plt
import numpy as np
import kikuchipy as kp

s

WARNING:hyperspy.api:The ipywidgets GUI elements are not available, probably because the hyperspy_gui_ipywidgets package is not installed.
WARNING:hyperspy.api:The traitsui GUI elements are not available, probably because the hyperspy_gui_traitsui package is not installed.

[1]:

<EBSD, title: patterns My awes0m4 ..., dimensions: (3, 3|60, 60)>


Then we can define a detector with the same parameters as the one used to acquire the small Nickel data set

[2]:

detector = kp.detectors.EBSDDetector(
shape=s.axes_manager.signal_shape[::-1],
pc=[0.421, 0.779, 0.505],
convention="tsl",
px_size=70,  # microns
binning=8,
tilt=0,
sample_tilt=70
)
detector

[2]:

EBSDDetector (60, 60), px_size 70 um, binning 8, tilt 0, azimuthal 0, pc (0.421, 0.221, 0.505)

[3]:

detector.pc_tsl()

[3]:

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 EBSDDetector.pc_emsoft() and similar. Conversions between conventions are implemented as described in . 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 EBSDDetector.plot() (see its docstring for a complete explanation of its parameters)

[4]:

detector.plot(pattern=s.inav[0, 0].data)


This will produce a figure similar to the left panel in the detector coordinates figure above, without the arrows and colored labels.

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)

[5]:

detector.pc = np.ones([3, 4, 3]) * [0.421, 0.779, 0.505]

[5]:

(3, 4)

[6]:

detector.navigation_dimension

[6]:

2

[7]:

detector.pc = detector.pc[0, 0]

[7]:

(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 the detector coordinates figure above shows the detector plotted in the gnomonic projection using EBSDDetector.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

[8]:

detector.bounds

[8]:

array([ 0, 59,  0, 59])

[9]:

detector.gnomonic_bounds

[9]:

array([[-0.83366337,  1.14653465, -0.43762376,  1.54257426]])

[10]:

detector.x_range

[10]:

array([[-0.83366337,  1.14653465]])

[11]:

detector.r_max  # Largest radial distance to PC

[11]:

array([[1.92199819]])


## Projection center calibration¶

The gnomonic projection (pattern) center (PC) of an EBSD detector can be estimated by the “moving-screen” technique [HH91]. The technique relies on the assumption that the beam normal, shown in the top figure (d) above, is normal to the detector screen as well as the incoming electron beam, and will therefore intersect the screen at a position independent of the detector distance (DD). To find this position, we need two EBSD patterns acquired with a stationary beam but with a known difference $$\Delta z$$ in DD, say 5 mm.

First, the goal is to find the pattern position which does not shift between the two camera positions, ($$PC_x$$, $$PC_y$$). This point can be estimated in fractions of screen width and height, respectively, by selecting the same pattern features in both patterns. The two points of each pattern feature can then be used to form a straight line, and two or more such lines should intersect at ($$PC_x$$, $$PC_y$$).

Second, the DD ($$PC_z$$) can be estimated from the same points. After finding the distances $$L_{in}$$ and $$L_{out}$$ between two points (features) in both patterns (in = operating position, out = 5 mm from operating position), the DD can be found from the relation

$\mathrm{DD} = \frac{\Delta z}{L_{out}/L_{in} - 1},$

where DD is given in the same unit as the known camera distance difference. If also the detector pixel size $$\delta$$ is known (e.g. 46 mm / 508 px), $$PC_z$$ can be given in the fraction of the detector screen height

$PC_z = \frac{\mathrm{DD}}{N_r \delta b},$

where $$N_r$$ is the number of detector rows and $$b$$ is the binning factor.

Let’s find an estimate of the PC from two single crystal Silicon EBSD patterns, which are included in the kikuchipy.data module

[12]:

s_in = kp.data.silicon_ebsd_moving_screen_in(allow_download=True)
s_in.remove_static_background()
s_in.remove_dynamic_background()

s_out5mm.remove_static_background()
s_out5mm.remove_dynamic_background()

Downloading file 'data/silicon_ebsd_moving_screen/si_in.h5' from 'https://github.com/pyxem/kikuchipy-data/raw/master/silicon_ebsd_moving_screen/si_in.h5' to '/home/docs/.cache/kikuchipy/0.5.2'.
100%|████████████████████████████████████████| 493k/493k [00:00<00:00, 515MB/s]

Removing the static background:
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Removing the dynamic background:
[########################################] | 100% Completed |  0.1s

Downloading file 'data/silicon_ebsd_moving_screen/si_out5mm.h5' from 'https://github.com/pyxem/kikuchipy-data/raw/master/silicon_ebsd_moving_screen/si_out5mm.h5' to '/home/docs/.cache/kikuchipy/0.5.2'.
100%|████████████████████████████████████████| 493k/493k [00:00<00:00, 525MB/s]

Removing the static background:
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Removing the dynamic background:
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As a first approximation, we can find the detector pixel positions of the same features in both patterns by plotting them and noting the upper right coordianates provided by Matplotlib when plotting with an interactive backend (e.g. qt5 or notebook) and hovering over image pixels

[13]:

fig, ax = plt.subplots(ncols=2, sharex=True, sharey=True, figsize=(20, 10))
ax[0].imshow(s_in.data, cmap="gray")
_ = ax[1].imshow(s_out5mm.data, cmap="gray")


For this example we choose the positions of three zone axes. The PC calibration is performed by creating an instance of the PCCalibrationMovingScreen class

[14]:

cal = kp.detectors.PCCalibrationMovingScreen(
pattern_in=s_in.data,
pattern_out=s_out5mm.data,
points_in=[(109, 131), (390, 139), (246, 232)],
points_out=[(77, 146), (424, 156), (246, 269)],
delta_z=5,
px_size=None,  # Default
convention="tsl",  # Default
)
cal

[14]:

PCCalibrationMovingScreen: (PCx, PCy, PCz) = (0.5123, 0.8606, 21.6518)
3 points:
[[[109 131]
[390 139]
[246 232]]

[[ 77 146]
[424 156]
[246 269]]]


We see that ($$PC_x$$, $$PC_y$$) = (0.5123, 0.8606), while DD = 21.7 mm. To get $$PC_z$$ in fractions of detector height, we have to provide the detector pixel size $$\delta$$ upon initialization, or set it directly and recalculate the PC

[15]:

cal.px_size = 46 / 508  # mm/px
cal

[15]:

PCCalibrationMovingScreen: (PCx, PCy, PCz) = (0.5123, 0.8606, 0.4981)
3 points:
[[[109 131]
[390 139]
[246 232]]

[[ 77 146]
[424 156]
[246 269]]]


We can visualize the estimation by using the (opinionated) convenience method PCCalibrationMovingScreen.plot()

[16]:

cal.plot()


As expected, the three lines in the right figure meet at a more or less the same position. We can replot the three images and zoom in on the PC to see how close they are to each other. We will use two standard deviations of all $$PC_x$$ estimates as the axis limits (scaled with pattern shape)

[17]:

# PCy defined from top to bottom, otherwise "tsl", defined from bottom to top
cal.convention = "bruker"
pcx, pcy, _ = cal.pc
two_std = 2 * np.std(cal.pcx_all, axis=0)

fig, ax = cal.plot(return_fig_ax=True)
ax[2].set_xlim([cal.ncols * (pcx - two_std), cal.ncols * (pcx + two_std)])
_ = ax[2].set_ylim([cal.nrows * ( pcy - two_std), cal.nrows * (pcy + two_std)])


Finally, we can use this PC estimate along with the orientation of the Si crystal, as determined by Hough indexing with a commercial software, to see how good the estimate is, by performing a geometrical EBSD simulation of positions of Kikuchi band centres and zone axes from the five $$\{hkl\}$$ families $$\{111\}$$, $$\{200\}$$, $$\{220\}$$, $$\{222\}$$, and $$\{311\}$$

[18]:

from diffsims.crystallography import ReciprocalLatticePoint
from orix import crystal_map, quaternion

# Create simulation generator from a detector and crystal phase and orientation
detector = kp.detectors.EBSDDetector(
shape=cal.shape, pc=cal.pc, sample_tilt=70, convention=cal.convention
)
phase = crystal_map.Phase(space_group=227)
simgen = kp.generators.EBSDSimulationGenerator(
detector=detector, phase=phase, rotations=r
)

# Specify which plane families for which to simulate bands and zone axes
rlp = ReciprocalLatticePoint(
phase=phase, hkl=[[1, 1, 1], [2, 0, 0], [2, 2, 0], [2, 2, 2], [3, 1, 1]]
).symmetrise()  # Symmetrise to get all symmetrically equivalent planes
simgeo = simgen.geometrical_simulation(rlp)