E34 · Electron spectroscopy

EELS: Energy Loss & Thin-Specimen Signals

What do electrons lose when they pass through a thin specimen? Energy. They give it to plasmons, the collective oscillations of the valence electrons, and to tightly bound inner-shell electrons. The spectrum shows a zero-loss peak, a series of plasmon peaks and a core edge. Electrons in the zero-loss peak may still have scattered elastically. Repeated plasmon losses shift and smear the core edge, which is why the specimen has to be thin.

Move the probe across a carbon-rich inclusion and read two windows of the energy-loss spectrum from the transmitted electrons. Pick out the zero-loss peak, the repeated plasmon losses and the carbon core edge, then weigh collection angle, energy sampling, resolution and exposure before you trust the thin-specimen result.

Recorded-window log ratio $\text{—}$

Two or more low-loss events $0\,\mathrm{\%}$

Core counts minus known background $\text{—}$

Expected edge SNR · known background $\text{—}$

More measurements and target

Assumed relative thickness $\text{—}$

Expected low-loss window coverage $0\,\mathrm{\%}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

Specimen thickness

t

80\,\mathrm{nm}

Collection semi-angle

\beta

8\,\mathrm{mrad}

Energy response · FWHM

\Delta E

3\,\mathrm{eV}

Exposure per spectrum window

\tau

1\,\mathrm{s}

Advanced settings

Assumed low-loss mean free path

\lambda

100\,\mathrm{nm}

Detector dispersion

d_E

1\,\mathrm{eV/pixel}

Probe position

x

0\,\mathrm{nm}

Synthetic edge shift

\delta E

0\,\mathrm{eV}

Learning goal and model limits

Acquire a trustworthy thin spectrum

Resolve an edge without hiding plural loss

First predict how a thinner specimen changes repeated energy losses. With the probe on the carbon-rich region, reach a relative thickness of at most 0.35, an energy response of at most 1.5 eV, a dispersion of at most 0.5 eV per pixel and an expected edge SNR of at least 5. Then check the target.

Model assumptions

The beam is fixed at 200 kV and the probe current at 0.004 pA. The low-loss mean free path is an adjustable assumption, not a material calibration. Each loss is independent and centered at 16 eV with a 3 eV standard deviation, and the number of losses follows Poisson statistics. Low-loss collection is assumed not to depend on energy. The log ratio shown uses the actual counts from −10 to 130 eV and a finite zero-loss window, so broadening, noise and missing tails bias it. The core signal uses a separate approximation for rare core events: a synthetic carbon K edge near 284 eV is convolved with the low-loss distribution. Its yields, near-edge feature and shift do not identify a real chemical state. Core collection uses an assumed dipole angular distribution with negligible beam convergence. Net core counts subtract the known model background; they are not a fitted elemental analysis. The Gaussian instrument width is given as FWHM. Noise comes from a fixed seed: Poisson below 40 counts, a Gaussian approximation above. The magnetic sector disperses energy schematically at 0.1 scene unit per eV, and no magnetic field or aberrations are solved.

Start with the essentials

Focus question: What do electrons lose when they pass through a thin specimen?
One-sentence intuition: A zero-loss electron may still have scattered elastically. Repeated low losses shift and broaden the core edge, and the detector dispersion and the energy resolution set two different limits.

Core mathematical model

How many times an electron loses energy

P_n=\frac{\mu^n e^{-\mu}}{n!},\qquad \mu=\frac{t}{\lambda}

Independent inelastic events give a fraction of electrons with no loss and a population that lost energy more than once. The mean free path here is an assumed teaching input.

Relative thickness from counts

\left(\frac{t}{\lambda}\right)_{\!\mathrm{app}}=\ln\frac{I_{\mathrm{window}}}{I_{\mathrm{ZLP}}}

Both integrals use the recorded low-loss bins. Because the spectral window and the zero-loss integration are finite, the result can be biased, especially with a broad response or a thick specimen. An absolute thickness also needs a calibrated mean free path.

Instrument broadening and pixel integration

\sigma_E=\frac{\Delta E_{\mathrm{FWHM}}}{\sqrt{8\ln2}},\qquad C_j=N_{\mathrm{inc}}\int_{E_j^-}^{E_j^+}p(E)\,\mathrm dE

Resolution and dispersion are different things: narrower detector bins cannot undo an energy response that is already broad. Each bin integrates the probability first, and then counting noise is drawn.

A core event can be followed by low losses

S_{\mathrm{core}}(E)\propto\sum_{n=0}^{\infty}P_n\,[S_1*G_n](E-16n\,\mathrm{eV})

In this teaching approximation for rare core events, extra low losses move part of the edge toward higher loss. You can shift the synthetic onset and feature, but no real oxidation state is inferred. Subtracting the known model background gives an optimistic signal-to-noise reference.

Common difficulties

A shifted edge does not name a chemical state

Typical misconceptionAny shifted edge names a unique oxidation state.

Better mental modelThese near-edge shapes are invented. Identifying a chemical state needs reference spectra, calibration and a suitable scattering model.

Run the experiment

01
Start thin, then make it thicker
Compare Thin · resolved with Plural scattering. Watch the repeated low-loss peaks and compare the assumed thickness with the estimate from the recorded window.
What to observe: Multiple losses grow and spread the core signal to higher loss. A finite low-loss window can miss part of the tail.
02
Sampling is not resolution
Keep the specimen thin; compare 0.25 and 2 eV per pixel, then change the energy response from 1 to 6 eV.
What to observe: Fine bins preserve a narrow feature only if the instrument response is already narrow.
03
Locate the signal
Move the probe out of the carbon-rich region and back again. Then try the synthetic shift preset.
What to observe: The core edge weakens outside the inclusion. The low-loss average stays the same because the whole toy specimen has one mean free path.
04
Meet the target
Meet the stated limits on thickness, sampling, energy response and expected SNR, then check.
What to observe: The SNR with a known background is an optimistic reference. Real fitting, the detector response and material cross sections need further evidence.

Core mathematical model

How many times an electron loses energy

Relative thickness from counts

Instrument broadening and pixel integration

A core event can be followed by low losses

Common difficulties

A shifted edge does not name a chemical state

Run the experiment

Start thin, then make it thicker

Sampling is not resolution

Locate the signal

Meet the target