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title: "Aberration Corrector: Tune the Electron Probe Interactive Lab | Sandbox Physics"
description: "Can a wider cone of electrons focus to a sharper probe? Only if the leftover aberrations are tuned small."
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E31 · Electron wavefront

# Aberration Corrector: Tune the Electron Probe

**Can a wider cone of electrons focus to a sharper probe?** Only if the leftover aberrations are tuned small. A wider convergence angle can give a sharper probe, but the outer rays carry uncorrected wave aberrations that spread intensity out of the central peak. A Ronchigram that looks well tuned still does not prove the resolution you will get in an experiment.

Adjust the residual aberrations, each with its sign, record a Ronchigram, and compare it with the calculated probe at the specimen.

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

Model assumptions

This is a coherent 200 kV model with a thin phase screen as the specimen. It includes astigmatism and coma along fixed axes, signed defocus and third-order spherical aberration. The coefficients are residual aberrations, not corrector currents. Ronchigrams are simulated recorded diffraction counts. The probe intensity, aperture phase, Strehl ratio and encircled radius are calculated separately as references and are never reconstructed from the counts. A separate known-grating scan shares a fixed total electron count across five field positions and six frequencies. Its measured modulation and three-standard-error threshold define a sampled information limit and a usable field at 2.5 inverse nanometres. An assumed quadratic field-curvature law changes local focus; these are teaching calibration results, not microscope specifications. The model leaves out chromatic spread, source-size blur, higher-order aberrations, dose damage and material calibration. Angles and planes are magnified schematically.

Physics tutorial

## Aberration Corrector: Tune the Electron Probe

**Background** A corrected electron lens still has small residual aberrations, and they decide how sharp the focused probe can be. Here you tune those signed residuals, record a Ronchigram and compare it with the calculated probe at the specimen.

**Why it matters** A wider cone of electrons can focus to a smaller spot, but the residual aberrations grow with angle and can wash out the coherent peak.

Start with the essentials

- **Focus question**: Can a wider cone of electrons focus to a sharper probe?
- **One-sentence intuition**: The Ronchigram is measured in the diffraction plane and the probe is calculated in the specimen plane; compare the two, but do not mistake the calculation for a measurement.

### Core mathematical model

#### Wave aberration with signed coefficients

$$
\begin{aligned}\chi&=\frac{2\pi}{\lambda}\left[\frac{C_{10}}2\theta^2+\frac{C_{12}}2(\theta_x^2-\theta_y^2)\right.\\&\quad\left.+\frac{C_{21}}3\theta_x\theta^2+\frac{C_{30}}4\theta^4\right]\end{aligned}
$$

Angles are in radians and every coefficient is a length; spherical aberration is entered in micrometres. The sign of positive defocus follows this formula.

#### The probe and the recorded Ronchigram

$$
\begin{aligned}P&=\mathcal F^{-1}\!\left[Ae^{-i\chi}\right]\\I_R&=\left|\mathcal F\!\left[Pe^{i\phi}\right]\right|^2\end{aligned}
$$

A unitary discrete Fourier transform carries the normalized probe through one fixed, synthetic amorphous phase sheet.

#### Height of the coherent peak

$$
S=\frac{\left|\sum_{j\in A}e^{-i\chi_j}\right|^2}{N_A^2}
$$

This on-axis Strehl ratio compares the peak with that of the same aperture without aberrations. It is calculated, not inferred from the camera.

### Common difficulties

#### What a sharp calculated probe does not prove

**Typical misconception** A low residual or sharp calculated probe certifies an experimental image.

**Better mental model** The residuals and the calculated probe describe only this model. This is a coherent 200 kV model with a thin phase screen as the specimen. It includes astigmatism and coma along fixed axes, signed defocus and third-order spherical aberration. The coefficients are residual aberrations, not corrector currents. Ronchigrams are simulated recorded diffraction counts. The probe intensity, aperture phase, Strehl ratio and encircled radius are calculated separately as references and are never reconstructed from the counts. A separate known-grating scan shares a fixed total electron count across five field positions and six frequencies. Its measured modulation and three-standard-error threshold define a sampled information limit and a usable field at 2.5 inverse nanometres. An assumed quadratic field-curvature law changes local focus; these are teaching calibration results, not microscope specifications. The model leaves out chromatic spread, source-size blur, higher-order aberrations, dose damage and material calibration. Angles and planes are magnified schematically.

### Run the experiment

1. 01

   #### Predict what limits the probe

   Adjust residual aberrations and convergence. Predict how the probe changes.

   **What to observe:** A small aperture can hide aberrations, but it makes the probe wider.

2. 02

   #### Record a measurement

   Acquire a Ronchigram, then change a slider.

   **What to observe:** Each lens change records fresh counts and references; changing display gain preserves them.

3. 03

   #### Compare the two planes

   Inspect the measured diffraction plane and the separately calculated specimen-plane probe.

   **What to observe:** The aperture phase plot is a view into the model, not a second detector.

4. 04

   #### Measure the field at a fixed dose

   Compare the corrected, wide-field and low-electron presets. Open the recorded field and dose test.

   **What to observe:** The same three-standard-error rule tests every grating. A sharp reference probe can coexist with an undetectable modulation or a poor field edge.

5. 05

   #### Check the target, then test the limits

   Check the target, then compare astigmatism, spherical aberration and low-count cases.

   **What to observe:** Correcting the lens and collecting enough counts are two separate requirements.

Physical conclusion

Finishing the teaching task does not establish an experimental resolution or a unique phase recovery.

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)