Restrict to fundamental sector

This example shows how to restrict the stereographic plot to the fundamental sector of one of the eleven Laue group symmetries \(S\) using restrict_to_sector(). The sector is typically obtained from orix.quaternion.Symmetry.fundamental_sector. It is often called the ‘fundamental triangle’ or ‘standard stereographic triangle (SST)’.

We demonstrate this functionality by drawing (near) great circles about some typically strongly reflecting low-index reciprocal lattice vectors \(\mathbf{g} = \{hkl\}\) in crystals of point group \(S = m\bar{3}m\). The deviations from the great circles are related to the kinematically calculated width of a Kikuchi band scattered from these vectors, assuming a lattice parameter of $a$ = 0.404 nm (aluminium) and an accelerating voltage of 20 kV. The band width is assumed to be two times the Bragg angle \(\theta\).

Miller (4,), point group m-3m, hkl
[[1. 1. 1.]
 [2. 0. 0.]
 [2. 2. 0.]
 [3. 1. 1.]]

import matplotlib.pyplot as plt
import numpy as np

from orix import plot
from orix.crystal_map import Phase
from orix.quaternion import symmetry
from orix.vector import Miller

plt.rcParams["font.size"] = 15

# Symmetrically equivalent set of hkl
g0 = Miller(
    hkl=[[1, 1, 1], [2, 0, 0], [2, 2, 0], [3, 1, 1]],
    phase=Phase(point_group=symmetry.Oh),
)
print(g0)
g, idx = g0.symmetrise(unique=True, return_index=True)

# Width of Kikuchi bands (deviation from great circles)
theta = np.deg2rad([1.054, 1.218, 1.722, 2.019])
theta = theta[idx]

# Plot pair of near great circles
fig = g.draw_circle(opening_angle=np.pi / 2 + theta, return_figure=True)
g.draw_circle(opening_angle=np.pi / 2 - theta, figure=fig)

# Restrict to fundamental sector of m-3m (with some padding outside sector)
ax = fig.axes[0]
ax.restrict_to_sector(
    g.phase.point_group.fundamental_sector, edgecolor="r", lw=2, pad=5
)

# Get symmetrically equivalent set of zone axes t = <uvw>
t = g.reshape(g.size, 1).cross(g.reshape(1, g.size)).flatten()
t = t.in_fundamental_sector()
t = t.unique(use_symmetry=True)
t = t.round().unique()
t.scatter(
    figure=fig,
    c="none",
    vector_labels=plot.format_labels(t.coordinates),
    text_kwargs={"va": "center", "bbox": {"fc": "w", "pad": 1, "alpha": 0.75}},
)

_ = ax.set_title(r"Low-index $[uvw]$ in fundamental sector of $m\bar{3}m$", pad=10)

Estimated memory usage: 9 MB