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Sandbox Physics

E32 · Angular selection and column contrast

STEM Detectors: Choose the Scattering Angle

Move the collection bounds through bright field and three annular geometries, using the same stored counts to test contrast and noise.

Interactive modelSTEM Detectors: Choose the Scattering Angle
Selected virtual detector counts—\text{—}
Selected sampled primary outcome total—\text{—}
Simulated counts beyond 250 mrad—\text{—}
Simulated removed-electron counts—\text{—}
Selected-point transmitted fraction—\text{—}
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{—}

Physics tutorial

STEM Detectors: Choose the Scattering Angle

BackgroundMove the collection bounds through bright field and three annular geometries, using the same stored counts to test contrast and noise.

Why it mattersTarget: inner angle at least twice the convergence semi-angle, heavier-column contrast at least 0.2, heavy/light ROI intensity ratio at least 1.5, pooled difference SNR at least 10 and assumed probe FWHM at most 0.15 nm.

Start with the essentials

Focus question
Which electrons make a heavier column brighter?
One-sentence intuition
Angle selection changes which stored electrons contribute to an image.

Core mathematical model

Count into angular bins

μj=Iτe pjNann=∑j∈ANj\begin{aligned}\mu_j&=\frac{I\tau}{e}\,p_j\\N_{\rm ann}&=\sum_{j\in\mathcal A}N_j\end{aligned}

Each virtual image sums stored radial counts in its declared half-open angular interval.

A declared angular envelope

f(θ)=θs2 e−θ2/(2s2)r=Ltan⁡θ\begin{aligned}f(\theta)&=\frac{\theta}{s^2}\,e^{-\theta^2/(2s^2)}\\r&=L\tan\theta\end{aligned}

The isotropic Gaussian radial law is an assumed transport surrogate. Radius and angle share the displayed collector geometry.

Audit two known regions

C=N‾2−N‾1N‾2+N‾1σΔ2=∑1Nn12+∑2Nn22S=∣N‾2−N‾1∣σΔ\begin{aligned}C&=\frac{\overline N_2-\overline N_1}{\overline N_2+\overline N_1}\\\sigma_\Delta^2&=\frac{\sum_1N}{n_1^2}+\frac{\sum_2N}{n_2^2}\\S&=\frac{|\overline N_2-\overline N_1|}{\sigma_\Delta}\end{aligned}

This pooled difference uses count statistics in separately declared regions; it is not a per-pixel SNR or composition inversion.

Common difficulties

A bright pixel is not a chemical assay

Typical misconceptionAny annular image directly identifies atoms.

Better mental modelThickness, angular acceptance and probe overlap also change contrast. Low-angle phase effects require a coherent model absent here.

Run the experiment

  1. 01

    Predict the contrast

    Compare the separate specimen input with bright field and the selected angular image.

    What to observe: Known regions are an audit, not a recovered chemical map.
  2. 02

    Change collection without reacquiring

    Change inner and outer angles while watching the native histogram and outcome budget.

    What to observe: Virtual images change; native radial counts stay fixed.
  3. 03

    Test a successful setting

    Use the resolved preset and check the target. Compare the count-derived row profiles.

    What to observe: Read the signed contrast and pooled difference SNR together.
  4. 04

    Find the limits

    Try low exposure, a broad probe and reversed bounds. Compare BF, ABF and low-angle ADF.

    What to observe: Gain changes appearance; it cannot recover missing counts or resolve a blurred probe.