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.

Empirical range proxy $0\,\mathrm{nm}$

Backscattered histories $0\,\mathrm{\%}$

Transmitted histories $0\,\mathrm{\%}$

Above-edge transfer · subset, not X-ray yield $0\,\mathrm{\%}$

More measurements and target

Stopped primary histories $0\,\mathrm{\%}$

SE 90-percent source depth $0\,\mathrm{nm}$

Auger 90-percent source depth $0\,\mathrm{nm}$

Assumed X-ray excitation edge $0\,\mathrm{keV}$

Incident energy transferred by primaries $0\,\mathrm{\%}$

Incident energy carried out by BSE $0\,\mathrm{\%}$

Incident energy carried out by transmission $0\,\mathrm{\%}$

Mean energy per transmitted primary $0\,\mathrm{keV}$

Experiment task $\text{—}$

Simulated examples; each plot identifies its data source.

Landing energy

E_0

10\,\mathrm{keV}

Atomic number

Z

14

Specimen thickness

t

2000\,\mathrm{nm}

Advanced settings

Secondary electronsShow illustrative routes; keep transport and budgets unchanged.Backscattered electronsShow illustrative routes; keep transport and budgets unchanged.Characteristic X-raysShow illustrative routes; keep transport and budgets unchanged.Auger electronsShow illustrative routes; keep transport and budgets unchanged.CathodoluminescenceShow illustrative routes; keep transport and budgets unchanged.Transmitted electronsShow illustrative routes; keep transport and budgets unchanged.Transmitted energy lossShow illustrative routes; keep transport and budgets unchanged.Mass density

\rho

2.33\,\mathrm{g\,cm^{-3}}

Learning goal and model limits

Instrument settings

Electron–Matter Signal Map

Target: at least 50 percent of the simulated primary electrons pass through, while 90 percent of the modeled secondary electrons come from less than 15 nm deep.

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?

BackgroundA 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 mattersBefore 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 misconceptionThe picture is a calibrated prediction for a real material.

Better mental modelThis 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 misconceptionThe transmitted primary percentage is the same as the transmitted energy percentage.

Better mental modelPrimaries 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

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.
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.
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.