Orientation mapping

Understanding crystal orientation

When you grow or process crystalline materials, individual grains form with different orientations. Each grain has the same crystal structure (same lattice, same atoms), but rotated differently in space. This orientation affects how the material responds to stress, conducts electricity, and interacts with light. Mapping these orientations reveals grain boundaries, texture, and processing history.

What is orientation mapping?: Orientation mapping determines how each crystal grain is rotated relative to a reference frame. In 4D-STEM, we analyze diffraction patterns at every scan position to identify the crystal’s orientation at that location. By measuring many positions, we create a map showing how orientation varies across the sample.
Why do we need to know the crystal structure first?: Orientation is meaningless without a reference. You cannot say “this grain is rotated 45°” unless you know what structure you are rotating. The lattice parameters, symmetry, and atomic basis define the reference frame. Without knowing the phase (crystal structure), you have nothing to orient.
How does the algorithm work?: Colin Ophus’s approach matches the measured diffraction pattern against a library of simulated patterns from all possible crystal orientations. The best match identifies the local orientation. By repeating this at every scan position, you build a complete orientation map.
What makes this challenging?: Crystal symmetry creates ambiguity. A cubic crystal looks identical from multiple orientations due to symmetry operations. The algorithm must account for these equivalent orientations. Additionally, overlapping grains, crystal defects, and noise complicate pattern matching.
What do we learn from orientation maps?: Orientation maps reveal grain boundaries (where orientation changes abruptly), crystallographic texture (preferred orientations from processing), twin boundaries (specific orientation relationships), deformation patterns (orientation gradients from strain), and phase boundaries (where different crystal structures meet).

Simulating diffraction patterns

Algorithm for computing diffraction patterns

We simulate diffraction patterns for FCC gold (lattice parameter \(a = 4.078\) Å) using the metric tensor to construct the reciprocal lattice, the Laue condition to identify allowed reflections, followed by structure factor calculations to determine their intensities.

Construct reciprocal lattice from known crystal structure

In practice, orientation mapping is performed on a sample whose phase and crystal structure are already known (identified by XRD, selected area diffraction, or prior characterization). We start with the lattice parameters from the known structure and construct the reciprocal lattice (see crystallography section for details on the metric tensor and reciprocal lattice construction).

For gold FCC with lattice parameter \(a = 4.078\) Å, the reciprocal lattice parameter is:

\[a^* = \frac{2\pi}{a} \approx 1.540 \text{ Å}^{-1}\]

Any reciprocal lattice vector \(\mathbf{g}_{hkl}\) has magnitude:

\[|\mathbf{g}_{hkl}| = \frac{2\pi}{a}\sqrt{h^2 + k^2 + l^2}\]

Apply the Laue condition to identify allowed reflections

The Laue condition determines which reciprocal lattice points \(\mathbf{g}_{hkl}\) can contribute to the diffraction pattern:

\[\mathbf{g}_{hkl} \cdot \mathbf{u} = 0\]

where \(\mathbf{u}\) is the zone axis direction (beam direction). This ensures that the reciprocal lattice vector lies in the plane perpendicular to the electron beam.

For the [001] zone axis (\(\mathbf{u} = [0, 0, 1]\)):

\[\mathbf{g}_{hkl} \cdot [0, 0, 1] = l = 0\]

Only reflections with \(l = 0\) satisfy the Laue condition, restricting us to the \((h, k, 0)\) plane in reciprocal space.

For the [110] zone axis (\(\mathbf{u} = [1, 1, 0]/\sqrt{2}\)):

\[\mathbf{g}_{hkl} \cdot [1, 1, 0] = h + k = 0\]

Only reflections where \(h + k = 0\) appear, giving us reflections like \((2, -2, 0)\), \((1, -1, 3)\), etc.

Define atomic positions in the FCC unit cell

FCC gold has 4 atoms per unit cell at fractional coordinates:

\[\begin{split}\mathbf{r}_1 &= (0, 0, 0) \quad \text{(corner)} \\ \mathbf{r}_2 &= (1/2, 1/2, 0) \quad \text{(face center)} \\ \mathbf{r}_3 &= (1/2, 0, 1/2) \quad \text{(face center)} \\ \mathbf{r}_4 &= (0, 1/2, 1/2) \quad \text{(face center)}\end{split}\]

Calculate structure factors for allowed reflections

For each reflection \((h, k, l)\) that satisfies the Laue condition, compute the structure factor by summing phase contributions from all atoms:

\[F_{hkl} = \sum_{j=1}^{4} f_j \exp[2\pi i (h x_j + k y_j + l z_j)]\]

where \(f_j\) is the atomic scattering factor. For gold atoms (\(f_j = f\)), this becomes:

\[\begin{split}F_{hkl} &= f \left\{ 1 + \exp[\pi i(h+k)] + \exp[\pi i(h+l)] + \exp[\pi i(k+l)] \right\} \\ &= f \left\{ 1 + (-1)^{h+k} + (-1)^{h+l} + (-1)^{k+l} \right\}\end{split}\]

This reveals the FCC selection rule: \(F_{hkl} \neq 0\) only when \(h\), \(k\), and \(l\) are all even or all odd.

Compute diffraction intensities

The intensity of each allowed reflection is:

\[I_{hkl} = |F_{hkl}|^2\]

For example, in the [001] zone axis where \(l = 0\):

\((2, 2, 0)\): All even → \(F = 4f\), \(I = 16f^2\) (strong reflection)
\((2, 0, 0)\): Mixed parity → \(F = 0\), \(I = 0\) (forbidden, systematic absence)
\((4, 4, 0)\): All even → \(F = 4f\), \(I = 16f^2\) (strong reflection)

The reciprocal lattice positions are converted to physical coordinates using \(\mathbf{g}_{hkl} = h\mathbf{a}^* + k\mathbf{b}^* + l\mathbf{c}^*\) where \(\mathbf{a}^* = 2\pi/a \approx 1.540\) Å⁻¹ for gold.

By using the Laue condition first, we only compute intensities for reflections that can physically appear in the diffraction pattern. This is more efficient than calculating all possible \((h, k, l)\) combinations and eliminates reflections that violate the diffraction geometry. Each zone axis produces a unique fingerprint of allowed reflections, which enables orientation mapping by pattern matching.