---
title: "CBED: Thickness and Crystal Symmetry Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/cbed-thickness/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/cbed-thickness/"
last_modified: "2026-10-27"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## CBED: Thickness and Crystal Symmetry

**Background** A convergent beam tests many incident directions in one exposure. The intensity inside each disc carries information about thickness and about the three-dimensional crystal structure.

**Why it matters** Thickness and tilt both shape the intensity inside the discs. To measure thickness, you have to tell the two apart.

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 misconception** If I pass the task, I have solved a real structure from experiment.

**Better mental model** 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.

### Run the experiment

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

Physical conclusion

The two-beam fit estimates thickness and tilt; the HOLZ comparison holds independent thickness and tilt calibrations fixed. Both require competing solutions to be checked.

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)