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Sandbox Physics

E29 · Two-beam CBED

CBED: Read Thickness Inside the Disc

Record a convergent-beam pattern, fit its measured rocking profile, and inspect competing thickness solutions.

Interactive modelCBED: Read Thickness Inside the Disc
Stored frame count—\text{—}
Total recorded sensor counts—\text{—}
Recorded total incident dose—\text{—}
Selected recorded tilt—\text{—}
Count-derived thickness—\text{—}
Count-derived tilt—\text{—}
Competing separated thickness—\text{—}
Measured-fit relative RMS—\text{—}
Separated-candidate loss gap—\text{—}
Experiment target—\text{—}

Physics tutorial

CBED: Read Thickness Inside the Disc

BackgroundRecord a convergent-beam pattern, fit its measured rocking profile, and inspect competing thickness solutions.

Why it mattersCan one disc pair distinguish thickness from tilt?

Start with the essentials

Focus question
Can one disc pair distinguish thickness from tilt?
One-sentence intuition
More angular structure distinguishes thickness aliases.

Core mathematical model

Two-beam rocking intensity

ω=sgξgIg=sin⁡2 ⁣[πtξg1+ω2]1+ω2\begin{aligned}\omega&=s_g\xi_g\\I_g&=\frac{\sin^2\!\left[\frac{\pi t}{\xi_g}\sqrt{1+\omega^2}\right]}{1+\omega^2}\end{aligned}

Nonabsorbing coupled waves exchange intensity with thickness. The transmitted fraction is one minus this intensity.

Disc angle and excitation error

sg=g(β+αx)R=Lαd=Lλg\begin{aligned}s_g&=g(\beta+\alpha_x)\\R&=L\alpha\\d&=L\lambda g\end{aligned}

Tilt is relative to the chosen Bragg condition; angles in the equation are radians. Camera coordinates follow effective camera length.

Conditional count likelihood

ℓj=Cg,jln⁡Ig,j+C0,jln⁡(1−Ig,j)L=−∑jℓj\begin{aligned}\ell_j&=C_{g,j}\ln I_{g,j}\\&\quad+C_{0,j}\ln(1-I_{g,j})\\\mathcal L&=-\sum_j\ell_j\end{aligned}

Fit only measured paired-channel counts. Profile tilt at each thickness to show competing minima.

Common difficulties

Scientific boundary

Typical misconceptionA passed task is an experimentally solved structure.

Better mental modelNonabsorbing two-beam teaching crystal at 200 kV, assigned extinction distance 80 nm and reciprocal vector 2.5 inverse nanometres. Convergence stays below the nonoverlap limit. Counts cover equal incident angular cells over a fixed 100 square nanometre probe footprint; the footprint is held fixed rather than solving probe optics. Conditional count likelihood fits thickness and tilt on a finite grid. No HOLZ, point-group or polarity determination, overlapping-disc interference, absorption or calibrated material claim. Angles and specimen thickness are enlarged schematically.

Run the experiment

  1. 01

    Predict the angular fingerprint

    Choose foil thickness, convergence and tilt. Predict whether a wider disc can distinguish thickness aliases.

    What to observe: At fixed total dose, more angular information does not create more electrons.
  2. 02

    Acquire the disc pair

    Record both channels, then change an acquisition slider.

    What to observe: Stored camera counts stay unchanged until acquisition is committed again.
  3. 03

    Fit and inspect ambiguity

    Fit thickness and tilt; compare the measured rocking curve with the fitted curve and the profiled loss.

    What to observe: A second separated minimum can explain a narrow disc almost as well.
  4. 04

    Test an adverse case

    Compare the narrow-disc and low-count completed examples, then check the target.

    What to observe: Likelihood separation is a teaching diagnostic, not a calibrated confidence interval.