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# Geometrical EBSD simulations¶

This section details how to inspect and visualize the results from pattern matching or Hough indexing of cubic crystals by plotting Kikuchi bands and zone axes onto an EBSD signal. We consider this a geometrical EBSD simulation, since it's only the Kikuchi band centres and zone axis positions that will be computed. These simulations are based on the work by Aimo Winkelmann in the supplementary material to Britton et al. (2016).

Let's import the necessary libraries and a small (3, 3) Nickel EBSD test data set

In [ ]:
# Exchange inline for notebook or qt5 (from pyqt) for interactive plotting
%matplotlib inline

import tempfile
from diffsims.crystallography import ReciprocalLatticePoint
import hyperspy.api as hs
import matplotlib
import matplotlib.pyplot as plt
import numpy as np
from orix.crystal_map import Phase
from orix.quaternion import Rotation
import kikuchipy as kp

plt.rcParams.update({
"figure.figsize": (10, 10), "font.size": 20, "lines.markersize": 10
})

s


Let's enhance the Kikuchi bands by removing the static and dynamic backgrounds

In [ ]:
s.remove_static_background()
s.remove_dynamic_background()

In [ ]:
_ = hs.plot.plot_images(
s, axes_decor=None, label=None, colorbar=False, tight_layout=True
)


To project Kikuchi bands and zone axis positions onto our detector, we need

1. a description of the crystal phase, in the geometrical case only the space group
2. the set of Kikuchi bands to consider, e.g. the {111}, {200}, {220}, and {311} crystal plane families
3. the crystal orientations with respect to the reference frame
4. the position of the detector with respect to the sample reference frame, in the form of a sample-detector model which includes the sample and detector tilt and the projection center (shortes distance from the source point on the sample to the detector), given here as (PC$_x$, PC$_y$, PC$_z$)

We'll store the crystal phase information in an orix.crystal_map.Phase instance

In [ ]:
phase = Phase(name="ni", space_group=225)
phase


We'll set up the relevant Kikuchi bands (and the zone axes from these) in a diffsims.crystallography.ReciprocalLatticePoint instance

In [ ]:
rlp = ReciprocalLatticePoint(
phase=phase, hkl=[[1, 1, 1], [2, 0, 0], [2, 2, 0], [3, 1, 1]]
)
rlp


Let's plot where the $(hkl)$ plane normals impinge on the unit sphere by showing their stereographic projection on the upper and lower hemispheres

In [ ]:
scatter_kwargs = dict(grid=True, hemisphere="both", axes_labels=["x", "y", None])
colors = [f"C{i}" for i in range(rlp.size)]
rlp.hkl.scatter(
c=colors,
vector_labels=[str(hi).replace("[", "(").replace("]", ")") for hi in rlp.hkl.data],
**scatter_kwargs
)


We can get a new instance with the symmetrically equivalent planes $\{hkl\}$ in each plane family using ReciprocalLatticePoint.symmetrise()

In [ ]:
rlp2, mult = rlp.symmetrise(return_multiplicity=True)

# Plot {111} plane family
fig = rlp2.hkl[:mult[0]].scatter(c=colors[0], return_figure=True, **scatter_kwargs)

# Plot remaining plane families
i = 0
for j in range(mult.size):
rlp2.hkl[i:i + mult[j]].scatter(c=colors[j], figure=fig)
i += mult[j]


We know from pattern matching of these nine patterns to dynamically simulated patterns of orientations uniformly distributed in the orientation space of the proper point group $432$, that they come from two grains with orientations of about $(\phi_1, \Phi, \phi_2) = (80^{\circ}, 34^{\circ}, -90^{\circ})$ and $(\phi_1, \Phi, \phi_2) = (115^{\circ}, 27^{\circ}, -95^{\circ})$. We store these orientations in an orix.quaternion.Rotation instance

In [ ]:
grain1 = (80, 34, -90)
grain2 = (115, 27, -95)
rot = Rotation.from_euler(
[grain1, grain2, grain2],
[grain1, grain2, grain2],
[grain1, grain2, grain2]
]),
)
rot


We describe the sample-detector model in an kikuchipy.detectors.EBSDDetector instance. From Hough indexing we know the projection center to be, in the EDAX TSL convention (see the reference frame guide for the various conventions and more details on the use of the sample-detector model), $(x^{*}, y^{*}, z^{*}) = (0.421, 0.7794, 0.5049)$. The sample was tilted $70^{\circ}$ about the microscope X direction towards the detector, and the detector normal was orthogonal to the optical axis (beam direction)

In [ ]:
detector = kp.detectors.EBSDDetector(
shape=s.axes_manager.signal_shape[::-1],
sample_tilt=70,
pc=[0.421, 0.7794, 0.5049],
convention="tsl"
)
detector


Note that the projection center gets converted internally to the Bruker convention.

Let's create an EBSDSimulationGenerator instance

In [ ]:
sim_gen = kp.generators.EBSDSimulationGenerator(
detector=detector,
phase=phase,
rotations=rot
)
sim_gen


Now we're ready to simulate geometrical EBSD patterns from the generator and the sets of crystal plane families

In [ ]:
sim = sim_gen.geometrical_simulation(reciprocal_lattice_point=rlp2)
sim


We see that 27 of the 50 Kikuchi bands we're visible on the detector in the nine patterns, stored in an instance of the kikuchipy.simulations.features.KikuchiBand class, which is a class inheriting from the ReciprocalLatticePoint.

In [ ]:
sim.zone_axes.size


We also see that there are 91 zone axes resulting from the 27 Kikuchi bands, stored in an instance of the kikuchipy.simulations.features.ZoneAxis class, also inheriting from ReciprocalLatticePoint.

We can now add these simulations as markers to be displayed on top of our experimental EBSD signal when plotting

In [ ]:
markers = sim.as_markers(pc=False)


The markers update with the pattern when navigating, thus helping us determine whether an indexing was successful, and in labeling the bands and zone axes in the pattern

In [ ]:
s.plot(navigator=None)


Whether to plot only bands, zone axes, zone axes labels, projection center, or all of them, can be set in the GeometricalEBSDSimulation.as_markers() method. Their appearance on the pattern can also be controlled to some extent. The above method itself calls bands_as_markers(), pc_as_markers(), zone_axes_as_markers(), and zone_axes_labels_as_markers(). See their documentation for available modifications.

Let's first remove the markers from the signal, and add only the bands and zone axes

In [ ]:
del s.metadata.Markers
marker=sim.as_markers(
pc=False,
zone_axes_labels=False,
bands_kwargs=dict(
family_colors=["w", "magenta", "cyan", "lime"], linestyle="--",
),
zone_axes_kwargs=dict(
marker="s", size=150, facecolor="none", edgecolor="w",
),
),
plot_marker=False,
permanent=True,
)

In [ ]:
s.plot(navigator=None)

In [ ]:
del s.metadata.Markers


We can write single EBSD patterns with the markers on top to file

In [ ]:
fig = s._plot.signal_plot.figure
bbox = matplotlib.transforms.Bbox.from_extents(
np.array(fig.axes[0].bbox.extents) / 72  # The denominator may vary
)

In [ ]:
nav_x, nav_y = s.axes_manager.indices
temp_dir = tempfile.mkdtemp() + "/"
fname = temp_dir + f"geosim_y{nav_y}_x{nav_x}.png"
s._plot.signal_plot.figure.savefig(fname, bbox_inches=bbox, dpi=150)

In [ ]:
s.axes_manager.indices = (2, 1)
s.plot(navigator=None, colorbar=False, axes_off=True, title="")

In [ ]:
nav_x, nav_y = s.axes_manager.indices
fname = temp_dir + f"geosim_y{nav_y}_x{nav_x}.png"
s._plot.signal_plot.figure.savefig(fname, bbox_inches=bbox, dpi=150)


The coordinates of these bands and zone axes are available as class attributes. For the bands, we can e.g. extract the plane trace coordinates (y0, x0, y1, x1) in either gnomonic or detector coordinates (taking into account the detector size in pixels) for all bands or per navigation position

In [ ]:
sim.bands[0, 0].plane_trace_coordinates[:10]  # Gnomonic

In [ ]:
sim.bands_detector_coordinates[0, 0, :10]  # Detector


The NaN values signify that that particular band is not visible on the detector in that position. The crystal plane normal of each band, pointing from the source point to the detector, is also available

In [ ]:
sim.bands[1, 1].hkl_detector[:10]


And where the vector hits the detector, in either detector or gnomonic coordinates

In [ ]:
sim.bands[0, 0].x_detector

In [ ]:
sim.bands[0, 0].x_gnomonic


The same information is available for the zone axes

In [ ]:
sim.zone_axes_label_detector_coordinates[0, 0][20:30]  # Detector

In [ ]:
sim.zone_axes[0, 1].uvw_detector[:10]

In [ ]:
sim.zone_axes[0, 0].y_detector[:10]

In [ ]:
sim.zone_axes[0, 0].y_gnomonic[:10]


With this information, we can go around kikuchipy when plotting and only use matplotlib

In [ ]:
nav_idx = (2, 1)[::-1]

fig, ax = plt.subplots(figsize=(5, 5))
ax.imshow(s.inav[nav_idx], cmap="gray")

print(sim.bands_detector_coordinates.shape)
for i in np.ndindex(sim.bands_detector_coordinates.shape[2]):
sim_slice = nav_idx + (i,)
coords = sim.bands_detector_coordinates[sim_slice][0]
y0, x0, y1, x1 = coords
ax.axline((y0, x0), (y1, x1), linestyle="--", color="w")

print(sim.zone_axes_detector_coordinates.shape)
for j in np.ndindex(sim.zone_axes_detector_coordinates.shape[2]):
sim_slice = nav_idx + (j,)
coords = sim.zone_axes_detector_coordinates[sim_slice][0]
x, y = coords
ax.scatter(x=x, y=y, zorder=5, s=50)

_ = ax.axis((0, 59, 59, 0))
ax.axis("off");

In [ ]:
# Remove files written to disk in this user guide
import os
for file in ["geosim_y0_x0.png", "geosim_y1_x2.png"]:
os.remove(temp_dir + file)
os.rmdir(temp_dir)