---
title: "Electron–Matter Signal Map Interactive Lab | Sandbox Physics"
description: "When an electron beam enters a specimen, where do the different signals come from? From different depths. Secondary electrons have so little energy that only those made in a thin layer near the surface can escape. Backscattered electrons, beam electrons bounced back out, come from deeper. X-rays come from wherever the electrons still carry enough energy to excite them. Beam energy, atomic number and density set these depths."
canonical_url: "https://sandboxphysics.com/labs/electron-signal-map/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/electron-signal-map/"
last_modified: "2026-10-25"
---

E05 · Electron microscopy / specimen physics

# Electron–Matter Signal Map

**When an electron beam enters a specimen, where do the different signals come from?** From different depths. Secondary electrons have so little energy that only those made in a thin layer near the surface can escape. Backscattered electrons, beam electrons bounced back out, come from deeper. X-rays come from wherever the electrons still carry enough energy to excite them. Beam energy, atomic number and density set these depths.

Change the beam energy, atomic number, density and thickness of the specimen, and follow simulated primary electrons along reproducible, seeded random paths through a cutaway view. Turn signal routes on and off, and compare where each signal comes from, where the primaries end up and how the conserved incident energy is shared out. The challenge is to transmit at least half the primary beam while secondary electrons still come only from near the surface.

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

Model assumptions

This is a teaching Monte Carlo. The empirical Kanaya–Okayama range sets the total path length for continuous slowing down, and scattering follows an invented random-walk rule; no measured cross sections are used. The signal branches are schematic, not calibrated yields. Secondary and Auger escape lengths are assumed, and cathodoluminescence requires a luminescent material. The X-ray proxy counts energy transferred locally while the primary is above the edge; it does not count photons. Transferred energy includes secondary channels that are not tracked, so it is not all heat. Exit fractions ignore detector acceptance. Secondary and Auger electrons are assumed to be generated uniformly, and only within the depth the primaries can reach. The model cannot do quantitative composition analysis.

Physics tutorial

## When an electron beam enters a specimen, where do the different signals come from?

**Background** A beam electron entering a solid scatters and slows down along a random path, and different signals come from different depths along the way. Change the beam energy, atomic number, density and thickness, follow the seeded paths in the cutaway sample, and turn signal routes on and off to compare their source depths. Try to transmit at least half the primary beam while keeping secondary electrons surface sensitive.

**Why it matters** Before you read an electron image, it helps to know which features come from physics in the specimen and which come from how the detector shows them.

Start with the essentials

- **Focus question**: When an electron beam enters a specimen, where do the different signals come from?
- **One-sentence intuition**: Counting how many primaries escape is not the same as measuring how much energy leaves: an escaping primary carries out the energy it still has, and only the energy it lost counts as transferred.

### Core mathematical model

#### Empirical electron range

$$
R=\frac{27.6AE_0^{1.67}}{\rho Z^{0.889}}\,\mathrm{nm}
$$

Energy is in keV and density in grams per cubic centimetre. The atomic mass is the tabulated value for C, Si and Au, and a stated approximation for other elements. The low-energy end is an extrapolation.

#### How much of each signal escapes

$$
w(z)\propto g(z)\exp(-z/\ell_{\mathrm{esc}})
$$

Secondary and Auger electrons are assumed to be generated uniformly down to the range or the thickness, whichever is smaller, and to escape over fixed lengths. Every curve is scaled to its own peak. The X-ray proxy weights the energy the primaries transfer locally above the edge by an assumed attenuation. None of the curves is an absolute yield.

#### Continuous slowing down

$$
E(s)=E_0\left(1-\frac{s}{R}\right)^{1/1.67}
$$

The range is the total length of the path, not how deep the electron gets. Each segment is split where it crosses a depth-bin boundary, and its energy there comes from the remaining range.

#### Where the incident energy goes

$$
\begin{gathered} f_{\mathrm{tr}}+f_{\mathrm{BSE}}+f_{\mathrm{T}}=1\\ f_{\mathrm{tr}}=\frac{\sum\Delta E}{NE_0}\end{gathered}
$$

All three fractions are shares of the total incident primary energy. Transferred energy includes excitations and secondary radiation that are not tracked, not only heat. Energy carried out by escaping electrons is not deposited.

#### Energy that can excite X-rays

$$
\begin{gathered}\Delta E_{>}=(E_a-E_c)_+-(E_b-E_c)_+\\(x)_+=\max(x,0)\end{gathered}
$$

Only energy lost while the primary is still above the assumed shell edge goes into the X-ray proxy. It is part of the transferred energy, not an extra term in the energy balance, and it is not a photon-production efficiency.

### Common difficulties

#### Limits of the model

**Typical misconception** The picture is a calibrated prediction for a real material.

**Better mental model** This is a teaching Monte Carlo. The empirical Kanaya–Okayama range sets the total path length for continuous slowing down, and scattering follows an invented random-walk rule; no measured cross sections are used. The signal branches are schematic, not calibrated yields. Secondary and Auger escape lengths are assumed, and cathodoluminescence requires a luminescent material. X-ray excitation uses a rough shell threshold. The model cannot do quantitative composition analysis.

#### Counts versus energy

**Typical misconception** The transmitted primary percentage is the same as the transmitted energy percentage.

**Better mental model** Primaries slow down before they escape. The transmitted energy fraction equals the transmitted count fraction times their mean exit energy divided by the landing energy. Neither number includes detector collection efficiency.

### Run the experiment

1. 01

   #### Predict

   Compare the Silicon block and Transmitting foil presets. Look at both the share of primaries transmitted and the energy they carry out.

   **What to observe:** A signal produced deep inside the specimen can be absorbed before it reaches a detector. Transmission changes with thickness, but surface signals still escape only from shallow depths.

2. 02

   #### Tune toward the target

   Adjust the controls until at least 50 percent of the primary histories pass through and 90 percent of the modeled secondary electrons come from less than 15 nm deep.

   **What to observe:** The target is judged by the numerical readouts, not by how bright the picture looks.

3. 03

   #### Explain the limit

   Compare the gold block below and above its assumed shell edge, then hide each signal route in turn. Explain why the transport results stay the same while the X-ray proxy disappears below the threshold.

   **What to observe:** The routes are drawn on top of a teaching Monte Carlo. The empirical Kanaya–Okayama range sets the total path length for continuous slowing down, and scattering follows an invented random-walk rule; no measured cross sections are used. The signal branches are schematic, not calibrated yields. Secondary and Auger escape lengths are assumed, and cathodoluminescence requires a luminescent material. X-ray excitation uses a rough shell threshold. The model cannot do quantitative composition analysis.

Physical conclusion

Count fractions and energy fractions are different quantities, and an X-ray signal also needs the primary to have enough energy at that spot.

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)