E06 · Electron microscopy / specimen physics

Electron Scattering & Mean Free Paths

How thin must a specimen be to avoid multiple scattering? Thin compared with the inelastic mean free path, the average distance an electron travels before it loses energy. Then most transmitted electrons lose energy once at most. The log of the ratio of all transmitted electrons to those that lost none estimates the thickness, but an energy range that is cut short or too few detected electrons skews it.

Record simulated electron scattering through a foil. Compare the angular pattern of all electrons with that of the zero-loss electrons, read the saved energy-loss spectrum, and see how the collection angle and a truncated energy window change the apparent thickness ratio.

Recorded in-window electrons $0$

Recorded zero loss / incident monitor $0\,\mathrm{\%}$

Count-derived apparent log ratio $0$

Approximate counting standard error $0$

More measurements and target

Simulated incident monitor counts $0$

Recorded zero-loss electrons $0$

Count adequacy · excludes window bias $\text{—}$

Separate assumed inelastic thickness ratio $0$

Predicted spectral window coverage $0\,\mathrm{\%}$

Assumed elastic mean free path $0\,\mathrm{nm}$

Assumed inelastic mean free path $0\,\mathrm{nm}$

Predicted zero-loss fraction · no aperture $0\,\mathrm{\%}$

Predicted no-event fraction $0\,\mathrm{\%}$

Recorded plural fraction · within window $0\,\mathrm{\%}$

Projected RMS angle · direct plus diffuse $0\,\mathrm{mrad}$

Predicted collected zero-loss fraction $0\,\mathrm{\%}$

Predicted mean loss · full spectrum $0\,\mathrm{eV}$

Experiment task $\text{—}$

Simulated examples; each plot identifies its data source.

Beam energy

E_0

100\,\mathrm{keV}

Specimen thickness

t

80\,\mathrm{nm}

Collection semi-angle

\beta

20\,\mathrm{mrad}

Expected incident electrons

N_{\mathrm{in}}

20000

Advanced settings

Aligned crystal proxyReduce the illustrative elastic scattering rate.Atomic number

Z

14

Mass density

\rho

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

Recorded maximum loss

E_{\max}

240\,\mathrm{eV}

Log image display gain only

G

1

Learning goal and model limits

Instrument settings

Electron Scattering & Mean Free Paths

Target: record at least 200 counts in the window and at least 100 zero-loss counts. Keep the thickness at 20 nm or more, the recorded zero-loss collection at 60 percent or more, the recorded plural-loss fraction below 10 percent and the predicted window coverage at 99 percent or more.

Model assumptions

This teaching model treats angular scattering and energy-loss events as independent, with Poisson-distributed losses, assumed mean free paths and a fixed 16 eV loss per event. The detector is an ideal counter of angle and loss together; analyzer fields, real coupling between angle and loss, and material calibration are left out. The aligned-crystal option only changes an assumed scattering rate. A ratio from a finite energy window is an apparent ratio, and its counting error does not include the bias from the window.

Physics tutorial

When does a zero-loss spectrum give a useful thickness ratio?

BackgroundEELS estimates foil thickness from the ratio of all transmitted electrons to those that lost no energy. Record a simulated acquisition, compare the zero-loss electrons with all recorded electrons, and read an apparent inelastic thickness ratio from the saved spectrum.

Why it mattersThe bright central spot can still contain electrons that were deflected elastically. And if energy-loss counts are missing from the spectrum, a thick specimen can look thinner than it is.

Start with the essentials

Focus question: Can the same specimen give a different log ratio when the energy window is shortened?
One-sentence intuition: Zero loss does not mean unscattered, and the collection angle drops out of the ideal full-spectrum ratio only because this model treats angle and loss as independent.

Core mathematical model

Independent inelastic events

P_n=\frac{\mu^n e^{-\mu}}{n!},\quad\mu=\frac{t}{\ell_i}

Each event loses a fixed 16 eV. The mean free paths and the aligned-crystal switch are illustrative; there are no material cross sections and no dynamical diffraction.

Apparent ratio from the counts

\widehat{\mu}_{\mathrm{app}}=\ln\!\left(\frac{I_W}{I_0}\right)

The numerator is the sum of the recorded counts inside the window, and the denominator is the zero-loss part of them. The estimate uses no specimen input and no ideal spectrum. If no zero-loss counts were recorded, the ratio is unavailable; no pseudocount is added.

Bias from a truncated window

\mu_{\mathrm{app}}\longrightarrow\mu+\ln C_W,\quad C_W=\sum_{n=0}^{m}P_n

With many counts the apparent ratio approaches this limit, which falls below the assumed inelastic thickness ratio whenever the window misses some losses. More exposure reduces counting noise but does not remove that bias.

Approximate counting error

\widehat{\sigma}_{\mu}=\sqrt{\frac{1}{I_0}-\frac{1}{I_W}}

The zero-loss counts are part of the window counts, so the two are correlated. This delta-method standard error appears only when there are at least 20 zero-loss counts and some energy-loss counts. It is not a confidence guarantee, and it does not bound the errors from spectral truncation, the model or material calibration.

Common difficulties

An aperture is not an energy filter

Typical misconceptionAll electrons in the central angular spot must be zero loss.

Better mental modelIn this model an electron can lose energy without being deflected elastically, so it stays in the central spot. One angular pixel can hold both zero-loss and inelastic counts.

What the simulated detector leaves out

Typical misconceptionThe apparatus is a real detector that measures angle and energy together, or a calibrated EELS analyzer.

Better mental modelThe detector is idealized: it tags each electron's angle exactly and counts how many losses it had. Analyzer dispersion, finite energy resolution, real angle-loss coupling, detector efficiency, saturation, coherent diffraction and material calibration are left out. The drawn rays lie in one effective central scattering plane, with enlarged angles.

Run the experiment

01
Predict, then acquire
Load the Thin foil and Plural scattering presets. Compare the saved spectrum, the zero-loss counts and the separate assumed thickness ratio.
What to observe: The two angular maps and the spectrum are built from the same recorded electrons. The zero-loss map is a subset of them, not a separately generated image.
02
Change settings without touching the data
First change only the display gain. Then change the Collection semi-angle or Specimen thickness, and press Acquire counts to apply the new settings.
What to observe: Until you acquire again, new settings leave the record, the apparatus view and the readouts unchanged. Gain only changes the logarithmic display of the angular maps.
03
Find the bias from a short window
Load the Truncated spectrum preset. Raise the Recorded maximum loss, acquire again, and compare the apparent ratio and the coverage.
What to observe: A reasonable-looking number from the counts can still be biased by the missing losses. A very thick specimen may give no zero-loss counts at all, and then the ratio is unavailable.
04
Complete the task
Reach the count and collection targets with a suitably thin foil and an energy window that covers nearly the whole spectrum.
What to observe: The thresholds are teaching targets. Having enough counts does not by itself make the thickness reliable for a real material.