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# Kikuchi pattern simulations¶

This section explains how to inspect and visualize the results from EBSD indexing by plotting Kikuchi lines and zone axes onto an EBSD signal. We consider this a geometrical EBSD simulation, since it's only positions of Kikuchi lines and zone axes that are computed. These simulations are based on the work by Aimo Winkelmann in the supplementary material to Britton et al. (2016).

We'll also show how to perform kinematical Kikuchi pattern simulations.

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 diffpy.structure import Atom, Lattice, Structure
from diffsims.crystallography import ReciprocalLatticeVector
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
import pyvista

# Plotting parameters
plt.rcParams.update(
{"figure.figsize": (10, 10), "font.size": 20, "lines.markersize": 10}
)
pyvista.global_theme.window_size = [700, 700]
pyvista.set_jupyter_backend("pythreejs")

s


## Geometrical simulations¶

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 lines and zone axes onto our detector, we need

1. a description of the crystal phase

2. the set of Kikuchi bands to consider, e.g. the sets of planes {111}, {200}, {220}, and {311}

3. the crystal orientations with respect to the reference frame

4. the position of the detector with respect to the sample, 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(
space_group=225,
structure=Structure(
atoms=[Atom("Ni", [0, 0, 0])], lattice=Lattice(3.52, 3.52, 3.52, 90, 90, 90)
),
)

print(phase)
print(phase.structure)


We'll build up the reflector list using diffsims.crystallography.ReciprocalLatticeVector

In [ ]:
ref = ReciprocalLatticeVector(
phase=phase, hkl=[[1, 1, 1], [2, 0, 0], [2, 2, 0], [3, 1, 1]]
)
ref


We'll obtain the symmetrically equivalent vectors and plot each family of vectors in a distinct colour in the stereographic projection

In [ ]:
ref = ref.symmetrise().unique()
ref.size

In [ ]:
ref.print_table()

In [ ]:
# Dictionary with {hkl} as key and indices into ref as values
hkl_sets = ref.get_hkl_sets()
hkl_sets

In [ ]:
hkl_colors = np.zeros((ref.size, 3))
for idx, color in zip(
hkl_sets.values(),
[[1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 0]],  # Red, green, blue, yellow
):
hkl_colors[idx] = color

In [ ]:
hkl_labels = []
for hkl in ref.hkl.round(0).astype(int):
hkl_labels.append(str(hkl).replace("[", "(").replace("]", ")"))

In [ ]:
ref.scatter(c=hkl_colors, grid=True, ec="k", vector_labels=hkl_labels)


We can also plot the plane traces, i.e. the Kikuchi lines, in both hemispheres (they are identical for Ni)

In [ ]:
ref.draw_circle(
color=hkl_colors, hemisphere="both", figure_kwargs=dict(figsize=(15, 10))
)


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 = np.deg2rad((80, 34, -90))
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="edax",
)
detector


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

Now we're ready to create geometrical simulations. We create simulations using the kikuchipy.simulations.KikuchiPatternSimulator, which takes the reflectors as input

In [ ]:
simulator = kp.simulations.KikuchiPatternSimulator(ref)

In [ ]:
sim = simulator.on_detector(detector, rot)


By passing the detector and crystal orientations to KikuchiPatternSimulator.on_detector(), we've obtained a kikuchipy.simulations.GeometricalKikuchiPatternSimulation, which stores the detector and gnomonic coordinates of the Kikuchi lines and zone axes for each crystal orientation

In [ ]:
sim


We see that not all 50 of the reflectors in the reflector list are present in some pattern.

These geometrical simulations can be plotted one-by-one by themselves

In [ ]:
sim.plot()


Or, they be plotted on top of patterns in three ways: passing a pattern to GeometricalKikuchiPatternSimulation.plot()

In [ ]:
sim.plot(index=(1, 2), pattern=s.inav[2, 1].data)


Or obtaining a collection of lines, zone axes and zone axes labels as Matplotlib objects via GeometricalKikuchiPatternSimulation.as_collections() and adding them to an existing Matplotlib axis

In [ ]:
fig, ax = plt.subplots(ncols=3, nrows=3, figsize=(15, 15))

ax[idx].imshow(s.data[idx], cmap="gray")
ax[idx].axis("off")

lines, zone_axes, zone_axes_labels = sim.as_collections(
idx,
zone_axes=True,
zone_axes_labels=True,
zone_axes_labels_kwargs=dict(fontsize=12),
)
for label in zone_axes_labels:

fig.tight_layout()


Or obtaining the lines, zone axes, zone axes labels and PCs as HyperSpy markers via GeometricalKikuchiPatternSimulation.as_markers() and adding them to a signal of the same navigation shape as the simulation instance. This enables navigating the patterns with the geometrical simulations

In [ ]:
markers = sim.as_markers()

# To delete previously added permanent markers, do


In [ ]:
s.plot()


## Kinematical simulations¶

We can obtain kinematical master patterns using KikuchiPatternSimulator.calculate_master_pattern(), provided that the simulator is created from a ReciprocalLatticeVector instance that satisfy these conditions:

1. The unit cell, i.e. the structure used to create the phase used in ReciprocalLatticeVector, must have all asymmetric atom positions filled, which can either be done by creating a Phase instance from a valid CIF file with Phase.from_cif() or calling ReciprocalLatticeVector.sanitise_phase()

2. The atoms in the structure have their elements described by the symbol (Ni), not by the atomic number (28)

3. The lattice parameters are in given in Ångström.

4. Kinematical structure factors $F_{hkl}$ have been calculated with ReciprocalLatticeVector.calculate_structure_factor()

5. Bragg angles $\theta_B$ have been calculated with ReciprocalLatticeVector.calculate_theta()

Let's simulate three master patterns:

• nickel

• variant of the $\sigma$-phase (Fe, Cr) in steels

• silicon carbide 6H.

### Nickel¶

We'll compare our kinematical simulations to dynamical simulations performed with EMsoft (see Callahan and De Graef (2013)), since we have a Ni master pattern available in the kikuchipy.data module

In [ ]:
mp_ni_dyn = kp.data.nickel_ebsd_master_pattern_small(projection="stereographic")


Inspect phase

In [ ]:
phase_ni = mp_ni_dyn.phase.deepcopy()

print(phase_ni)
print(phase_ni.structure.lattice)


Change lattice parameters from nm to Ångström

In [ ]:
lat_ni = phase_ni.structure.lattice  # Shallow copy
lat_ni.setLatPar(lat_ni.a * 10, lat_ni.b * 10, lat_ni.c * 10)

print(phase_ni.structure.lattice)


We'll build up the reflector list by:

1. Finding all reflectors with a minimal interplanar spacing $d$

2. Keeping those that have a structure factor above 0.5% of the reflector with the highest structure factor

In [ ]:
ref_ni = ReciprocalLatticeVector.from_min_dspacing(phase_ni, 0.5)

ref_ni = ref_ni[
ref_ni.allowed
]  # Exclude non-allowed reflectors (not available for hexagonal or trigonal phases!)
ref_ni = ref_ni.unique(use_symmetry=True).symmetrise()


Sanitise phase

In [ ]:
ref_ni.phase.structure

In [ ]:
ref_ni.sanitise_phase()
ref_ni.phase.structure


We can now calculate the structure factors. Two parametrizations are available, from Kirkland (1998) ("xtables", the default) or Lobato and Van Dyck (2014) ("lobato")

In [ ]:
ref_ni.calculate_structure_factor()

In [ ]:
structure_factor_ni = abs(ref_ni.structure_factor)
ref_ni = ref_ni[structure_factor_ni > 0.05 * structure_factor_ni.max()]

ref_ni.print_table()

In [ ]:
ref_ni.calculate_theta(20e3)


We can now create our simulator and plot the simulation

In [ ]:
simulator_ni = kp.simulations.KikuchiPatternSimulator(ref_ni)
simulator_ni.reflectors.size


Plotting the band centers with intensities scaled by the structure factor

In [ ]:
simulator_ni.plot()


Or no scaling (scaling="square" for the structure factor squared)

In [ ]:
simulator_ni.plot(scaling=None)


We can also plot the Kikuchi bands, showing both hemispheres, also adding the crystal axes alignment

In [ ]:
fig = simulator_ni.plot(hemisphere="both", mode="bands", return_figure=True)

ax = fig.axes[0]
ax.scatter(simulator_ni.phase.a_axis, c="r")
ax.scatter(simulator_ni.phase.b_axis, c="g")
ax.scatter(simulator_ni.phase.c_axis, c="b")


The simulation can be plotted in spherical projection as well using Matplotlib, or PyVista provided that it is installed

In [ ]:
simulator_ni.plot("spherical", mode="bands")

In [ ]:
# Intensity scaling is not available when plotting in a notebook
simulator_ni.plot("spherical", mode="bands", backend="pyvista")  # Interactive!


When we're happy with the reflector list in the simulator, we can generate our kinematical master pattern

In [ ]:
mp_ni_kin = simulator_ni.calculate_master_pattern(half_size=200)


The returned master pattern is an instance of EBSDMasterPattern in the stereographic projection.

In [ ]:
mp_ni_kin

In [ ]:
mp_ni_kin.plot_spherical(style="points")  # Interactive!


Comparing kinematical and dynamical simulations

In [ ]:
# Exclude outside equator
ni_dyn_data = mp_ni_dyn.data.astype(np.float32)
ni_kin_data = mp_ni_kin.data.astype(np.float32)

fig, ax = plt.subplots(ncols=2)
ax[0].imshow(ni_kin_data, cmap="gray")
ax[1].imshow(ni_dyn_data, cmap="gray")
ax[0].axis("off")
ax[1].axis("off")
ax[0].set_title("Ni kinematical 20 kV")
ax[1].set_title("Ni dynamical 20 kV")
fig.tight_layout()

Warning Use dynamical simulations when performing pattern matching, not kinematical simulations. The latter intensities are not realistic, as demonstrated in the above comparison.

Finally, we can transform the master pattern in the stereographic projection to one in the Lambert projection

In [ ]:
mp_ni_kin_lp = mp_ni_kin.as_lambert()

In [ ]:
mp_ni_kin_lp.plot()


We can then project parts of this pattern onto our EBSD detector using get_patterns()

In [ ]:
s_kin = mp_ni_kin_lp.get_patterns(rot, detector, energy=20, compute=True)

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


Compare these to the ones the first plot!

### $\sigma$-phase¶

In [ ]:
phase_sigma = Phase(
name="sigma",
space_group=136,
structure=Structure(
atoms=[
Atom("Cr", [0, 0, 0], 0.5),
Atom("Fe", [0, 0, 0], 0.5),
Atom("Cr", [0.31773, 0.31773, 0], 0.5),
Atom("Fe", [0.31773, 0.31773, 0], 0.5),
Atom("Cr", [0.06609, 0.26067, 0], 0.5),
Atom("Fe", [0.06609, 0.26067, 0], 0.5),
Atom("Cr", [0.13122, 0.53651, 0], 0.5),
Atom("Fe", [0.13122, 0.53651, 0], 0.5),
],
lattice=Lattice(8.802, 8.802, 4.548, 90, 90, 90),
),
)
phase_sigma

In [ ]:
ref_sigma = ReciprocalLatticeVector.from_min_dspacing(phase_sigma, 1)

ref_sigma.sanitise_phase()

ref_sigma.calculate_structure_factor("lobato")

structure_factor = abs(ref_sigma.structure_factor)
ref_sigma = ref_sigma[structure_factor > 0.05 * structure_factor.max()]

ref_sigma.calculate_theta(20e3)

ref_sigma.print_table()

In [ ]:
simulator_sigma = kp.simulations.KikuchiPatternSimulator(ref_sigma)
simulator_sigma

In [ ]:
fig = simulator_sigma.plot(hemisphere="both", mode="bands", return_figure=True)

ax = fig.axes[0]
ax.scatter(simulator_sigma.phase.a_axis, c="r")
ax.scatter(simulator_sigma.phase.b_axis, c="g")
ax.scatter(simulator_sigma.phase.c_axis, c="b")
fig.tight_layout()

In [ ]:
simulator_sigma.plot("spherical", mode="bands", backend="pyvista")

In [ ]:
mp_sigma = simulator_sigma.calculate_master_pattern()

In [ ]:
mp_sigma.plot()

In [ ]:
mp_sigma.plot_spherical(style="points")  # Interactive!


### Silicon carbide 6H¶

In [ ]:
phase_sic = Phase(
name="sic_6h",
space_group=186,
structure=Structure(
atoms=[
Atom("Si", [1 / 3, 2 / 3, 0.20778]),
Atom("C", [1 / 3, 2 / 3, 0.33298]),
Atom("Si", [1 / 3, 2 / 3, 0.54134]),
Atom("C", [1 / 3, 2 / 3, 0.66647]),
Atom("C", [0, 0, 0]),
Atom("Si", [0, 0, 0.37461]),
],
lattice=Lattice(3.081, 3.081, 15.2101, 90, 90, 120),
),
)
phase_sic

In [ ]:
ref_sic = ReciprocalLatticeVector.from_min_dspacing(phase_sic)  # 0.7 Å, default
ref_sic.sanitise_phase()

ref_sic.calculate_structure_factor()

structure_factor = abs(ref_sic.structure_factor)
ref_sic = ref_sic[structure_factor > 0.05 * structure_factor.max()]

ref_sic.calculate_theta(20e3)

ref_sic.print_table()

In [ ]:
simulator_sic = kp.simulations.KikuchiPatternSimulator(ref_sic)
simulator_sic

In [ ]:
fig = simulator_sic.plot(hemisphere="both", mode="bands", return_figure=True)

ax = fig.axes[0]
ax.scatter(simulator_sic.phase.a_axis, c="r")
ax.scatter(simulator_sic.phase.b_axis, c="g")
ax.scatter(simulator_sic.phase.c_axis, c="b")

In [ ]:
simulator_sic.plot("spherical", mode="bands", backend="pyvista")

In [ ]:
mp_sic = simulator_sic.calculate_master_pattern(hemisphere="both", half_size=200)

In [ ]:
mp_sic

In [ ]:
mp_sic.plot(navigator=None)

In [ ]:
mp_sic.plot_spherical(style="points")  # Interactive!