---
title: "Electron Wavelength & Voltage Interactive Lab | Sandbox Physics"
description: "Is higher accelerating voltage always better? Only up to a point. Higher voltage shortens the wavelength, which helps resolution, but each electron also hits harder and can pass more energy to an atom. Past the knock-on threshold it knocks atoms out of their sites. Choose the highest voltage that stays below that threshold and still lets enough electrons through the specimen unscattered."
canonical_url: "https://sandboxphysics.com/labs/electron-wavelength/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/electron-wavelength/"
last_modified: "2026-10-25"
---

E01 · Electron microscopy / foundations

# Electron Wavelength & Voltage

**Is higher accelerating voltage always better?** Only up to a point. Higher voltage shortens the wavelength, which helps resolution, but each electron also hits harder and can pass more energy to an atom. Past the knock-on threshold it knocks atoms out of their sites. Choose the highest voltage that stays below that threshold and still lets enough electrons through the specimen unscattered.

Set the accelerating voltage and compare the classical and relativistic electron wavelengths. Watch how electrons scatter in a thin carbon foil and how much energy one electron can give a carbon atom. Find the shortest wavelength whose maximum energy transfer stays below the chosen knock-on threshold, with at least 40 percent of electrons passing through unscattered.

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

Model assumptions

Free-electron kinematics is exact and relativistic. The maximum energy transfer uses a heavy-nucleus approximation for carbon-12. The attenuation law and the normalized forward angular distribution are teaching assumptions: there are no measured cross sections, backscattering, absorption or Monte Carlo transport. The diffraction readout is the ideal scale for a circular aperture. Staying below the knock-on threshold does not rule out radiolysis, charging or heating.

Physics tutorial

## Is higher accelerating voltage always better?

**Background** Set the accelerating voltage and compare the classical and relativistic electron wavelengths. Watch how electrons scatter in a thin carbon foil and how much energy one electron can give a carbon atom. Find the shortest wavelength whose maximum energy transfer stays below the chosen knock-on threshold, with at least 40 percent of electrons passing through unscattered.

**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**: Is higher accelerating voltage always better?
- **One-sentence intuition**: A short wavelength and a surviving specimen are separate requirements. Staying below the knock-on threshold rules out knock-on events, but other kinds of damage can still happen.

### Core mathematical model

#### Relativistic wavelength

$$
\lambda=\frac{h}{\sqrt{2m_e eV\left(1+\frac{eV}{2m_e c^2}\right)}}
$$

Momentum, rather than the classical speed estimate, sets the matter-wave wavelength.

#### Ideal diffraction scale

$$
d_{\mathrm{diff}}=0.61\lambda/\alpha
$$

This does not include source size, aberrations, dose or detector sampling.

#### Heavy-nucleus energy transfer

$$
\begin{gathered}T_{\max}\simeq\frac{2K(K+2m_ec^2)}{Mc^2}\\K=eV\end{gathered}
$$

The target nucleus is always carbon-12. The threshold you set is an assumption; it is not a property shared by every carbon material.

#### Unscattered fraction in a foil

$$
\begin{gathered}P_0=\exp(-t/\ell)\\\ell=80\,\mathrm{nm}\left(\frac{V}{100\,\mathrm{kV}}\right)^{0.65}\end{gathered}
$$

The attenuation length and its voltage scaling are chosen for teaching comparisons; they are not a calibrated cross-section model.

#### Forward scattering angle distribution

$$
\begin{gathered}f(\theta)=\frac{\kappa e^{\kappa(\cos\theta-1)}}{2\pi(1-e^{-\kappa})}\\0\leq\theta\leq\pi/2,\quad\kappa=s^{-2}\end{gathered}
$$

This is a chosen distribution per steradian, symmetric about the beam axis and normalized over the forward hemisphere. Its scale is in radians and behaves like a transverse Gaussian width only at narrow angles. It is not a carbon cross section.

#### Probability of reaching the collector

$$
\begin{gathered}\beta=\arctan(R/L)\\C=P_0+(1-P_0)F(\beta)\\F(\beta)=2\pi\int_0^\beta f(\theta)\sin\theta\,d\theta\end{gathered}
$$

Every unscattered electron travels along the axis and lands on the centered ideal disk. The disk also catches some scattered electrons, so the unscattered share of the collected electrons is not the same as the unscattered fraction of the incident beam.

### Common difficulties

#### Damage below the knock-on threshold

**Typical misconception** A beam below the knock-on threshold is harmless.

**Better mental model** Radiolysis, heating, contamination and charging can still damage the specimen. Describing them needs additional material and dose models.

#### What a finite detector collects

**Typical misconception** The fraction collected by a finite detector equals the fraction that never scattered.

**Better mental model** A finite detector also catches some scattered electrons. In the density map, the central dot marks the unscattered beam, which is too narrow to resolve; the surrounding density, drawn on a fixed logarithmic scale, leaves that point out. The Lab does not simulate noise, detector efficiency or measured counts.

### Run the experiment

1. 01

   #### Predict the effect of voltage

   Predict how 300 kV changes wavelength and energy transfer relative to 80 kV, then compare the presets.

   **What to observe:** The wavelength gets shorter and the maximum energy transfer grows.

2. 02

   #### Approach the knock-on threshold

   Raise the maximum energy transfer to at least 85 percent of the assumed threshold without reaching it, and keep the unscattered fraction above 40 percent. Press Check target.

   **What to observe:** You are pushing for a short wavelength while respecting two limits set by the specimen: one on energy transfer and one on the unscattered fraction.

3. 03

   #### Test the classical approximation

   Turn off the relativistic model at high voltage, then increase foil thickness.

   **What to observe:** The classical wavelength comes out longer than the relativistic value. A thicker foil leaves fewer electrons unscattered.

4. 04

   #### Move the collector

   Move the collector farther away. Compare the collected probability, the missed probability and the unscattered share. Then change the convergence semi-angle.

   **What to observe:** The collected probability falls, while the wavelength, the foil attenuation and the knock-on comparison stay the same. The convergence angle changes only the separate ideal diffraction scale.

Physical conclusion

Higher voltage gives a shorter wavelength but also a larger maximum energy transfer to each atom, and staying below the knock-on threshold rules out only knock-on events.

Starter path · Step 3 of 6

## From a field to an electron microscope

[Next step → **Electron Diffraction & Reciprocal Space** Tilt a thin crystal, acquire a diffraction pattern and compare one spot's radius with the prediction. Then halve the camera length. Model · 20 min](https://sandboxphysics.com/labs/electron-diffraction/)

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)