E29 · Convergent-beam diffraction

CBED: Thickness and Crystal Symmetry

What can the intensity inside CBED discs reveal about a crystal? Each direction in the convergent beam probes how electron waves scatter through the crystal. Two-beam fringes can separate thickness from tilt. High-order reflections also respond to lattice strain, and their dynamical intensities can distinguish polar orientations within a specified family of structures and independent calibrations.

Measure thickness from two-beam fringes, then compare high-order diffraction lines from centrosymmetric and polar reference crystals. Watch how lattice strain moves the lines and how weak counts leave the crystal family unresolved.

Conditional reference family $\text{—}$

Best grid strain; conditional on calibration $\text{—}$

Family Poisson-deviance gap $\text{—}$

Poisson deviance per fitted bin $\text{—}$

More measurements and target

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 $\text{—}$

Simulated examples; each plot identifies its data source.

Convergence semi-angle

\alpha

1\,\mathrm{mrad}

Foil tilt relative to selected Bragg condition

\beta

0.6\,\mathrm{mrad}

Assigned crystal: 0 centrosymmetric, 1 forward, 2 reverse

1

Assigned horizontal lattice strain

\varepsilon_x

0\,\%

Foil thickness

t

64\,\mathrm{nm}

Total incident dose for next recording

D

50\,\frac{electrons}{nm^2}

Advanced settings

Model: 0 two beams, 1 HOLZ

0

Independent thickness calibration for HOLZ fit

t_c

64\,\mathrm{nm}

Effective camera length

L

1\,\mathrm{m}

Counts: 0 expectation, 1 seeded statistics

1

Data color gain only

G

1

Learning goal and model limits

Predict · record · fit

Read structure inside the discs

HOLZ target: distinguish the three oriented reference families by a Poisson-deviance gap of at least nine, with deviance per fitted bin at most two. The independent thickness and tilt calibrations are held fixed. In two-beam mode, resolve the competing thickness minima.

Sliders plan the next recording; stored counts stay fixed until acquisition. Drag the foil to tilt and acquire in CBED, or browse recorded angles in MicroED.

Model assumptions

Two separate teaching models share the camera. The two-beam model uses a reciprocal vector of 2.5 inverse nanometres and an 80 nm extinction distance. The HOLZ model propagates thirteen elastic waves through a fictional crystal with a 0.4 by 0.4 by 6 nm cell and assigned Gaussian scattering factors. Its three oriented reference families have unstrained point groups 4/mmm or 4mm; horizontal strain lowers those symmetries. A finite candidate search estimates strain and compares the centrosymmetric family with the two polar orientations, holding independently calibrated thickness and tilt fixed. A wrong calibration, an unknown structure, absorption or omitted reflections can invalidate that comparison. It is not a general point-group determination. Each angular cell collects the same incident exposure over a fixed 100 square nanometre footprint; the probe optics are not solved. The discs do not overlap. The full field resamples the same stored cells for display. Angles and foil thickness are drawn enlarged.

Start with the essentials

Focus question: What can the intensity inside CBED discs reveal about a crystal?
One-sentence intuition: The more of the rocking curve a disc covers, the easier it is to tell apart thicknesses that would otherwise fit equally well.

Core mathematical model

Elastic coupling and high-order excitation

\begin{aligned}i\,\partial_z\psi_g&=\sum_h H_{gh}\psi_h\\H_{gg}&=2\pi s_g\\s_g&=g_x\theta_x+g_y\theta_y+g_z\\&\quad+\tfrac{\lambda}{2}|\mathbf g|^2\end{aligned}

The Hermitian coupling matrix conserves elastic probability within the thirteen retained waves. Higher-order reflections have nonzero longitudinal reciprocal components. The strain changes the reciprocal lattice used by both the angular patterns and the camera geometry.

Diffracted intensity in the two-beam model

\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}

Without absorption, the transmitted and diffracted waves trade intensity back and forth as the crystal gets thicker. The transmitted fraction is one minus this intensity.

Disc angle and excitation error

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

Tilt is measured from the chosen Bragg condition, and angles in the equation are in radians. Positions on the camera scale with the effective camera length.

Likelihood of the recorded counts

\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}

Only the measured counts in the two paired channels are fitted. At each thickness the tilt is profiled out, which shows the other minima competing with the best one.

Common difficulties

Limits of the model

Typical misconceptionIf I pass the task, I have solved a real structure from experiment.

Better mental modelTwo separate teaching models share the camera. The two-beam model uses a reciprocal vector of 2.5 inverse nanometres and an 80 nm extinction distance. The HOLZ model propagates thirteen elastic waves through a fictional crystal with a 0.4 by 0.4 by 6 nm cell and assigned Gaussian scattering factors. Its three oriented reference families have unstrained point groups 4/mmm or 4mm; horizontal strain lowers those symmetries. A finite candidate search estimates strain and compares the centrosymmetric family with the two polar orientations, holding independently calibrated thickness and tilt fixed. A wrong calibration, an unknown structure, absorption or omitted reflections can invalidate that comparison. It is not a general point-group determination. Each angular cell collects the same incident exposure over a fixed 100 square nanometre footprint; the probe optics are not solved. The discs do not overlap. The full field resamples the same stored cells for display. Angles and foil thickness are drawn enlarged.

Run the experiment

01
Compare symmetry and polarity
Compare the polar, reverse-polar and centrosymmetric presets. Open the likelihood plot and inspect the competing strain curves.
What to observe: The oriented reference families produce different angular intensities through multiple elastic scattering. The decision uses the stored counts and independent calibration.
02
Move a high-order line
Change the horizontal lattice strain, acquire again and fit the stored counts. Compare the transmitted and first HOLZ discs.
What to observe: A lattice change shifts the reciprocal vectors and their excitation conditions. The estimate is restricted to a half-percent grid and the three supplied families.
03
Make the decision fail
Choose the low-count preset, or use a foil thickness that disagrees with the independent calibration. Acquire and fit again.
What to observe: A small gap leaves the family unresolved. A large deviance rejects all three candidates even when one beats the others. These cutoffs are teaching heuristics, not confidence intervals.
04
Predict what the disc will show
Choose the two-beam thickness preset, then set the foil thickness, convergence and tilt. Predict whether a wider disc can separate competing thicknesses.
What to observe: At a fixed total dose, a wider disc gives more angular information but no extra electrons.
05
Record the two discs
Record both channels, then change an acquisition slider.
What to observe: The stored camera counts do not change until you acquire again.
06
Fit and look for a second solution
Fit thickness and tilt. Compare the measured rocking curve with the fitted curve, and look at the profiled loss.
What to observe: For a narrow disc, a second, well-separated minimum can fit almost as well.
07
Try a hard case
Compare the finished examples with a narrow disc and with low counts, then check the target.
What to observe: The gap between likelihood minima is a teaching aid; it is not a calibrated confidence interval.

Core mathematical model

Elastic coupling and high-order excitation

Diffracted intensity in the two-beam model

Disc angle and excitation error

Likelihood of the recorded counts

Common difficulties

Limits of the model

Run the experiment

Compare symmetry and polarity

Move a high-order line

Make the decision fail

Predict what the disc will show

Record the two discs

Fit and look for a second solution

Try a hard case