Rotating z-vector to high-symmetry crystal directions

This example shows how to sample high-symmetry crystal directions \(\mathbf{t} = [u, v, w]\) (or zone axes) using orix.vector.Miller.from_highest_indices(). We will also return the rotations \(R\) which rotate \(\mathbf{v_z} = (0, 0, 1)\) to \(\mathbf{t}\).

We do the following to obtain the high-symmetry crystal directions:

1. Select a point group, here \(S = mmm\).

2. Sample all directions \(\mathbf{t_i}\) with indices of -1, 0, and 1

3. Project \(\mathbf{t_i}\) to the fundamental sector of the Laue group of \(S\) (which in this case is itself)

4. Discard symmetrically equivalent and other duplicate crystal directions. Vectors such as [001] and [002] are considered equal after we make them unit vectors.

5. Round the vector indices to the closest smallest integer (below a default of 20).

The rotations \(R\) can be useful e.g. when simulating diffraction patterns from crystals with one of the high-symmetry zone axes \(\mathbf{t}\) aligned along the beam path.

from orix.crystal_map import Phase
from orix.quaternion import Rotation
from orix.vector import Miller, Vector3d

phase = Phase(point_group="mmm")
t = Miller.from_highest_indices(phase, uvw=[1, 1, 1])
t = t.in_fundamental_sector()
t = t.unit.unique(use_symmetry=True).round()
print(t)

Miller (7,), point group mmm, uvw
[[0. 0. 1.]
 [0. 1. 1.]
 [0. 1. 0.]
 [1. 1. 1.]
 [1. 0. 1.]
 [1. 1. 0.]
 [1. 0. 0.]]

Get the rotations that rotate \(\mathbf{v_z}\) to these crystal directions

vz = Miller(uvw=[0, 0, 1], phase=t.phase)
R = Rotation.identity(t.size)
for i, t_i in enumerate(t):
    R[i] = Rotation.from_align_vectors(t_i, vz)
print(R)

Rotation (7,)
[[ 1.      0.      0.      0.    ]
 [ 0.9239 -0.3827  0.      0.    ]
 [ 0.7071 -0.7071  0.      0.    ]
 [ 0.8881 -0.3251  0.3251  0.    ]
 [ 0.9239  0.      0.3827  0.    ]
 [ 0.7071 -0.5     0.5     0.    ]
 [ 0.7071  0.      0.7071  0.    ]]

Plot the crystal directions within the fundamental sector of Laue group \(mmm\)

fig = t.scatter(
    vector_labels=[str(vi).replace(".", "") for vi in t.coordinates],
    text_kwargs={
        "size": 15,
        "offset": (0, 0.03),
        "bbox": {"fc": "w", "pad": 2, "alpha": 0.75},
    },
    return_figure=True,
)
fig.axes[0].restrict_to_sector(t.phase.point_group.fundamental_sector)

Estimated memory usage: 9 MB