---
title: "Electron Resolution & Aberration Budget Interactive Lab | Sandbox Physics"
description: "Once spherical aberration is corrected, what limits how small the probe can be? Whatever is now the biggest remaining contribution, such as chromatic aberration, the source size, drift or vibration. Removing one aberration also moves the best aperture size, and the next limit comes into view. Go through the remaining contributions one by one to find the new floor."
canonical_url: "https://sandboxphysics.com/labs/electron-aberrations/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/electron-aberrations/"
last_modified: "2026-10-25"
---

E04 · Electron microscopy / foundations

# Electron Resolution & Aberration Budget

**Once spherical aberration is corrected, what limits how small the probe can be?** Whatever is now the biggest remaining contribution, such as chromatic aberration, the source size, drift or vibration. Removing one aberration also moves the best aperture size, and the next limit comes into view. Go through the remaining contributions one by one to find the new floor.

Open or close the aperture and switch the spherical, chromatic, source, drift, vibration and astigmatism terms on and off. Compare three views: the radial phase, which the transfer curve also uses; a Gaussian estimate of the combined blur; and the weak-phase transfer cut off by the aperture. Pick a spatial frequency, then vary the energy width to tell chromatic damping apart from the aperture cutoff. Tune the aperture to within ten percent of the smallest probe these blur terms allow, then turn on correction and see what limit remains.

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

Model assumptions

The probe size comes from an engineering blur budget, drawn as a peak-normalized Gaussian; it is not a coherent point spread function. Correction leaves one percent of the spherical aberration plus a fixed 10 mm fifth-order term. A separate on-axis weak-phase reference uses the same semi-angle as a hard pupil cutoff. Its phase includes defocus, counted positive for underfocus, and third- and fifth-order spherical terms. Energy width is a Gaussian FWHM and sets the chromatic focal spread. Source size, drift and vibration enter as Gaussian image blurs given as FWHM. There is no acquired image, angular source coherence, astigmatic phase, lens instability, specimen or detector model. Ray errors and hardware are schematic, and the phase colors wrap around and are sampled for display.

Physics tutorial

## Once spherical aberration is corrected, what limits how small the probe can be?

**Background** Open or close the aperture and switch the spherical, chromatic, source, drift, vibration and astigmatism terms on and off. Compare the radial phase, the Gaussian reference for the blur budget and the weak-phase transfer cut off by the aperture. Tune the aperture to within ten percent of the best probe this budget allows, then turn on correction and see what limit remains.

**Why it matters** Before you trust an electron image, you need to know what the source, the lens column and the specimen itself allow.

Start with the essentials

- **Focus question**: Once spherical aberration is corrected, what limits how small the probe can be?
- **One-sentence intuition**: Correcting one aberration moves the best aperture and exposes the next largest blur term. The blur budget estimates probe size; the signed CTF describes how each spatial frequency gets through. They answer different imaging questions.

### Core mathematical model

#### Adding blur terms in quadrature

$$
\begin{aligned}d^2={}&d_{\mathrm{diff}}^2+d_{\mathrm{sph}}^2+d_{\mathrm{chr}}^2\\&+d_s^2+d_{\mathrm{drift}}^2+d_v^2+d_a^2+d_5^2\end{aligned}
$$

Each term is a characteristic diameter. Adding them in quadrature is an engineering approximation.

#### How aperture angle affects each term

$$
\begin{aligned}d_{\mathrm{diff}}&=0.61\lambda/\alpha\\d_{\mathrm{sph}}&=C_s\alpha^3/2\\d_{\mathrm{chr}}&=C_c\alpha\Delta E/E\end{aligned}
$$

A wider aperture angle shrinks the diffraction term and makes the aberration terms grow.

#### What correction leaves behind

$$
\begin{aligned}C_{s,\mathrm{res}}&=0.01C_s\\d_5&=C_5\alpha^5\\C_5&=10\,\mathrm{mm}\end{aligned}
$$

These residual values are for illustration; they do not describe the performance of any commercial corrector.

#### Weak-phase transfer function

$$
\begin{aligned}\chi(q)={}&-\pi\Delta f\lambda q^2\\&+\frac{\pi C_s\lambda^3q^4}{2}\\&+\frac{\pi C_5\lambda^5q^6}{3}\\T(q)&=-A(q)E_t(q)E_b(q)\sin\chi(q)\end{aligned}
$$

Underfocus counts as positive defocus. The pupil view shows the same radial phase. The hard pupil passes only frequencies inside the selected semi-angle. The fifth-order term appears only when spherical correction is on. This signed weak-phase reference is computed separately from the probe diameter budget and leaves out astigmatic phase.

#### Gaussian coherence envelopes

$$
\begin{aligned}\sigma_f&=\frac{C_c\Delta E}{E\sqrt{8\ln2}}\\E_t(q)&=\exp[-(\pi\lambda\sigma_f q^2)^2/2]\\E_b(q)&=\exp[-2\pi^2\sigma_b^2q^2]\end{aligned}
$$

The energy width is given as FWHM and converted to a standard deviation of focus. A separate image-blur standard deviation combines the source, drift and vibration FWHM values in quadrature. Neither envelope is a measured information limit.

### Common difficulties

#### What a corrector does not remove

**Typical misconception** Turning on a corrector eliminates every resolution limit.

**Better mental model** Source size, energy spread, drift, vibration and higher-order residuals remain as limits.

### Run the experiment

1. 01

   #### Find the best aperture

   Compare the Narrow aperture and Wide aperture presets and predict the best angle. Then tune to within ten percent of the best probe size and press Check target.

   **What to observe:** The best aperture lies in between, where diffraction and aberrations balance.

2. 02

   #### Find the remaining limit

   Turn on correction, then switch off the chromatic, drift and source terms one at a time.

   **What to observe:** Residual terms can limit the corrected probe even when spherical blur is small.

3. 03

   #### See contrast reversal

   Sweep defocus through positive and negative values. Choose a test frequency, close the aperture until that frequency is blocked, then reopen it and increase the energy width.

   **What to observe:** Defocus changes the phase but not the diameter budget. A frequency outside the aperture is transferred with exactly zero amplitude; chromatic damping lowers the envelope even inside the pupil.

Physical conclusion

Correcting one aberration moves the best aperture and exposes the next largest blur term. The blur budget estimates probe size; the signed CTF describes how each spatial frequency gets through. They answer different imaging questions.

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)