E32 · Coherent scanning and angular collection

STEM Detectors: Phase and Column Contrast

Why can changing the collection angles make the same atomic column bright or dark? Each angular region receives a different combination of electron waves. Inside the transmitted disc, their interference can make a column darker or reverse its contrast when focus changes. High-angle channels can show stronger peaks at heavier columns, but an intensity ratio also depends on the specimen and imaging conditions.

Compare bright field and annular channels formed from the same coherent scan. Change focus, thickness or lattice strain, and compare the measured intensities of light and heavy columns.

Selected virtual detector counts $\text{—}$

Light-column deficit relative to blank $\text{—}$

Light / blank difference SNR $\text{—}$

Heavy / light measured mean ratio $\text{—}$

More measurements and target

Selected sampled primary outcome total $\text{—}$

Simulated counts beyond 250 mrad $\text{—}$

Simulated removed-electron counts $\text{—}$

Selected-point transmitted fraction $\text{—}$

Empirical-model assumed probe FWHM $\text{—}$

Signed contrast in separate audit regions $\text{—}$

Pooled ROI difference SNR $\text{—}$

ROI 2 / ROI 1 mean count ratio $\text{—}$

Expected incident dose per scan pixel $\text{—}$

Experiment target $\text{—}$

Calculated half-energy probe diameter $\text{—}$

Simulated examples; each plot identifies its data source.

Projected slab thickness

t

8\,\mathrm{nm}

Exposure per scan pixel

\tau

10\,\mathrm{\mu s}

Inner collection semi-angle

\beta_{in}

0\,\mathrm{mrad}

Outer collection semi-angle

\beta_{out}

24\,\mathrm{mrad}

Signed probe defocus

\Delta f

0\,\mathrm{nm}

Horizontal lattice strain

\varepsilon_x

0\,\mathrm{\%}

Advanced settings

Probe convergence semi-angle

\alpha

25\,\mathrm{mrad}

Selected scan pixel index

k

264

Image display gain only

G

1

Model: 0 empirical, 1 coherent phase grating

0

Assigned light-column atomic-number proxy

Z_L

6

Learning goal and model limits

Predict · acquire · select · compare

Which scattering angles carry useful contrast?

Coherent target: collect the outer part of the illumination disc and distinguish the dark light-column region from the nearby blank by at least four statistical standard deviations. Coordinates are supplied; the measured heavy-to-light ratio is conditional on thickness, focus and collection angles. Empirical mode retains its original high-angle contrast task.

Model assumptions

The coherent branch is a fictional periodic phase grating at 200 keV. A 0.64 nm square cell contains two heavier columns and two lighter columns; assigned Gaussian potentials have width 0.015 nm and amplitudes proportional to an atomic-number proxy raised to 0.6. A normalized circular pupil forms the probe, with signed defocus phase. A single projected slab multiplies that probe before a unitary Fourier transform. There is no propagation between slices, channeling, thermal diffuse scattering, absorption, inelastic scattering or beam damage, so its high-angle channel is an elastic teaching example. A 32 by 32 scan records a 64 by 64 diffraction array at each point. Horizontal strain stretches the real-space period and changes reciprocal sampling consistently. Counts use a fixed seed, with a Gaussian approximation at high means. All virtual channels reuse those stored pixels; overlapping masks have correlated counts. Their color ranges are labelled separately. Known column regions and a nearby blank are supplied for conditional intensity and counting-noise comparisons, not elemental identification. The apparatus shows a representative axial point, with independently enlarged specimen and hardware. Its specimen color marks the prescribed projected potential. Empirical mode retains the earlier isotropic transport and Gaussian-probe example; its collection labels describe angle selection. No real material, instrument resolution or quantitative ABF/HAADF composition calibration is claimed.

Start with the essentials

Focus question: Why can changing the collection angles make the same atomic column bright or dark?
One-sentence intuition: A collection angle selects interfering waves as well as a number of electrons.

Core mathematical model

Probe and phase grating

\begin{aligned}P(\mathbf q)&=A(\mathbf q)e^{-i\pi\lambda\Delta f|\mathbf q|^2}\\\psi_{\mathbf R}(\mathbf r)&=\mathcal F^{-1}[P](\mathbf r-\mathbf R)\\T(\mathbf r)&=e^{i\sigma tV(\mathbf r)}\end{aligned}

The circular pupil is normalized to unit probability. Positive defocus uses the phase sign written here. Potential is in volts, slab thickness in nanometres and the interaction parameter in radians per volt nanometre.

Counts from one coherent diffraction record

\begin{aligned}\mu_{\mathbf R,\mathbf q}&=N_e|\mathcal F[T\psi_{\mathbf R}]|^2\\N_{\mathcal A}(\mathbf R)&=\sum_{\mathbf q\in\mathcal A}C_{\mathbf R,\mathbf q}\end{aligned}

At each known scan position, Fourier-pixel intensities form a count record. Every virtual image integrates a mask on that same record; a mask includes its lower edge and excludes its upper edge.

Light-column deficit and counting uncertainty

\begin{aligned}C_L&=\frac{\overline N_B-\overline N_L}{\overline N_B}\\\sigma_\Delta^2&=\frac{\sum_BN}{n_B^2}+\frac{\sum_LN}{n_L^2}\\S_L&=\frac{|\overline N_B-\overline N_L|}{\sigma_\Delta}\end{aligned}

The supplied light-column and nearby blank regions are disjoint. The difference uncertainty assumes independent count statistics; it is not a calibrated confidence interval or a composition estimate.

Real and reciprocal coordinates under strain

\begin{aligned}L_x&=L_0(1+\varepsilon_x)\\q_{x,m}&=m/L_x\\\theta_x&\simeq\lambda q_x\end{aligned}

Strain changes the physical field width, column spacing and horizontal diffraction-angle sampling. The image uses the corresponding physical aspect ratio.

Common difficulties

When a ratio can identify atoms

Typical misconceptionThe heavy-to-light count ratio is a universal atomic-number ruler.

Better mental modelThat ratio depends on the potential model, thickness, focus, probe and detector angles. This task reports the measured ratio for known regions and its statistical context. The coherent branch omits the thermal diffuse scattering and channeling needed for quantitative interpretation of many real STEM images. The coherent branch is a fictional periodic phase grating at 200 keV. A 0.64 nm square cell contains two heavier columns and two lighter columns; assigned Gaussian potentials have width 0.015 nm and amplitudes proportional to an atomic-number proxy raised to 0.6. A normalized circular pupil forms the probe, with signed defocus phase. A single projected slab multiplies that probe before a unitary Fourier transform. There is no propagation between slices, channeling, thermal diffuse scattering, absorption, inelastic scattering or beam damage, so its high-angle channel is an elastic teaching example. A 32 by 32 scan records a 64 by 64 diffraction array at each point. Horizontal strain stretches the real-space period and changes reciprocal sampling consistently. Counts use a fixed seed, with a Gaussian approximation at high means. All virtual channels reuse those stored pixels; overlapping masks have correlated counts. Their color ranges are labelled separately. Known column regions and a nearby blank are supplied for conditional intensity and counting-noise comparisons, not elemental identification. The apparatus shows a representative axial point, with independently enlarged specimen and hardware. Its specimen color marks the prescribed projected potential. Empirical mode retains the earlier isotropic transport and Gaussian-probe example; its collection labels describe angle selection. No real material, instrument resolution or quantitative ABF/HAADF composition calibration is claimed.

Run the experiment

01
Find the light columns
Start with the ABF light-column preset. Compare the selected channel with the high-angle image and read the light-to-blank difference.
What to observe: Both species give dark regions in this ABF example. In the high-angle elastic channel, their peaks have different strengths.
02
Reverse the phase contrast
Choose the phase-reversal preset, then move defocus and thickness.
What to observe: A light column can become brighter than the nearby blank even though its assigned species stays the same.
03
Select electrons from the same record
Drag the collector edge or move the collection angles. Expand the diffraction pixels and the other virtual images.
What to observe: Changing a virtual mask preserves the recorded Fourier pixels and their total count. Different masks can overlap.
04
Stretch the crystal and reduce counts
Compare the strain and low-exposure presets, then reset.
What to observe: Strain changes both the specimen width and reciprocal sampling. Low exposure can leave the light-column difference statistically unresolved.

Core mathematical model

Probe and phase grating

Counts from one coherent diffraction record

Light-column deficit and counting uncertainty

Real and reciprocal coordinates under strain

Common difficulties

When a ratio can identify atoms

Run the experiment

Find the light columns

Reverse the phase contrast

Select electrons from the same record

Stretch the crystal and reduce counts