E37 · Electron imaging

4D-STEM: One Scan, Many Virtual Images

Move the probe between two grains, inspect its recorded diffraction frame, then rotate a virtual detector after acquisition. Resolve the disks to recover orientation and lattice dilation from the same counts. Balance convergence, camera length, scan sampling, dose and data size.

Four data axes $\text{—}$

Stored count cube $\text{—}$

Selected acquired coordinate $\text{—}$

Selected frame · recorded counts $\text{—}$

Recovered orientation $\text{—}$

Recovered lattice dilation $\text{—}$

Expected detector coverage $\text{—}$

Valid lattice estimates $\text{—}$

Right-minus-left virtual contrast $\text{—}$

Incident scan dose $\text{—}$

Experiment target $\text{—}$

Acquire once, reprocess the counts

Isolate a grain without another scan

Use central bright-field maskOff: one movable first-order dark-field mask. Both sum the same recorded cube.Convergence semi-angle

\alpha

3.5\,\mathrm{mrad}

Effective camera length

L

0.3\,\mathrm{m}

Target scan step · grid rounds to fit

s

4\,\mathrm{nm}

Dwell per probe

\tau

100\,\mathrm{\mu s}

Right-grain orientation

\varphi

30\,\mathrm{deg}

Right-grain lattice dilation

\varepsilon

2\,\mathrm{\%}

Virtual dark-field mask azimuth

\varphi_M

0\,\mathrm{deg}

Virtual mask radius

\beta_M

1.8\,\mathrm{mrad}

Inspect probe · horizontal

x_p

16\,\mathrm{nm}

Inspect probe · vertical

y_p

0\,\mathrm{nm}

Predict which grain becomes bright when the virtual mask rotates. Resolve the lattice at more than 95% of probes, retain the full diffraction field, use a scan step no larger than 2 nm and dwell at least 200 microseconds, and obtain absolute grain contrast above 0.7 with a dark-field mask. Then check. Rotating the mask adds no dose.

Fixed 200 kV and 2 pA. An invented square lattice has 0.4 nm reference spacing, a straight grain boundary, adjustable right-grain rotation and isotropic dilation. The kinematic model assigns 60% to the direct disk and 10% to each of four first-order disks, with exact circle–pixel integration before counting. These yields are assumptions, not structure factors. A Gaussian probe-width approximation mixes grain intensities incoherently at the boundary; interference, multislice, dynamical scattering, higher orders, aberrations, drift, inelastic scattering, detector saturation and damage are omitted. The 12 mm detector has 48 square pixels per axis; effective camera length sets its angular field. The stored cube is uint32; changing a virtual mask or selected probe adds no exposure. A fourfold moment and four measured centroids estimate orientation modulo 90 degrees and scalar dilation; overlapping, clipped or low-count disks are masked gray. This is not general indexing or a strain-tensor solver. Data colors saturate at labeled display scales; numerical analysis uses linear counts. Changing acquisition parameters replaces a seeded realization, while animation adds no dose. Atomic marker spacing is enlarged by a factor of 35 relative to scan coordinates; markers illustrate grain orientation, not discrete atomic positions.

Start with the essentials

Focus question: What can one diffraction frame per probe position reveal?
One-sentence intuition: Changing a virtual aperture changes the sum of recorded pixels, not the electron exposure. Orientation and scalar dilation are inferred from measured disk positions, subject to separation, angular sampling and counting limits.

Core mathematical model

Four independent data axes

D_{ijmn}=D(x_i,y_j,\theta_{x,m},\theta_{y,n})

The first two axes locate the probe; the last two locate a detector pixel. The readout reports actual stored uint32 memory, excluding metadata and compression.

A virtual detector sums acquired counts

V_{ij}=\sum_{m,n}M_{mn}D_{ijmn},\qquad M_{mn}\in\{0,1\}

The green mask selects complete recorded detector pixels. A central mask gives bright field; a displaced mask selects one first-order disk. Mask boundaries are pixelated, not a continuous physical aperture.

Camera length converts angle to position

r=L\theta,\qquad \theta_B\simeq\frac{\lambda}{a}

The detector spans 12 mm. Increasing effective camera length improves angular sampling while reducing angular coverage. Here a square lattice is represented by the direct beam and four assumed first-order disks.

Scalar dilation from four Bragg centroids

\varepsilon=\frac{\theta_{B,\mathrm{ref}}}{\overline{\theta}_{B,\mathrm{meas}}}-1

An expanding lattice moves reciprocal-space disks inward. The fourfold angular moment estimates orientation modulo 90 degrees; four count-weighted centroids give a mean radius. This is an isotropic toy estimate, not a full strain tensor.

Dose and storage scale differently

\begin{gathered}N_e=\frac{I\tau}{e},\qquad d=\frac{N_eN_xN_y}{A}\\ S=4N_xN_yN_{\theta x}N_{\theta y}\ \mathrm{bytes}\end{gathered}

Finer scans add probe exposures and stored frames. Moving the inspection coordinate or virtual mask reuses the existing cube. Animation does not collect additional electrons.

Common difficulties

A cube is not a larger image

Typical misconceptionFour-dimensional STEM is one high-resolution two-dimensional picture.

Better mental modelTwo axes describe scanning and two describe scattering. Collapsing the detector axes with different masks produces different images from the same data.

Convergence is a tradeoff

Typical misconceptionA tighter real-space probe always gives more precise strain.

Better mental modelLarger convergence spreads reciprocal-space disks. The probe can become narrower while Bragg disks overlap and lattice-position inference fails.

Run the experiment

01
Predict which grain a mask selects
Start with Resolve · select left grain. Move the probe to each side and inspect the recorded frame.
What to observe: The disk pattern rotates and contracts in the right grain. Its orientation and dilation come from the selected counts.
02
Reuse the cube
Rotate the virtual dark-field mask from 0 to 30 degrees; compare with the right-grain preset and central bright-field mask.
What to observe: Dark-field contrast switches grain. The underlying diffraction frames, dose and memory stay fixed.
03
Break the inference deliberately
Compare overlapping disks, clipped diffraction and sparse low-count acquisition.
What to observe: Overlapping or truncated disk support invalidates lattice estimates. A visually bright partial frame does not restore missing information.
04
Complete the target
Find the stated coverage, valid fraction, scan step, dwell and dark-field contrast, then check.
What to observe: A passing target demonstrates this acquisition and estimator; it does not certify real material strain or orientation.