E34 · Electron spectroscopy

EELS: Energy Loss & Thin-Specimen Signals

Move a probe across a carbon-rich inclusion, collect transmitted electrons and read two spectrum windows. Separate the zero-loss peak, repeated plasmon losses and a carbon core edge. Balance collection, energy sampling, resolution and exposure before accepting a thin-specimen result.

Assumed relative thickness $\text{—}$

Recorded-window log ratio $\text{—}$

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

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

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

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

Experiment target $\text{—}$

Acquire a trustworthy thin spectrum

Resolve an edge without hiding plural loss

Specimen thickness

t

80\,\mathrm{nm}

Assumed low-loss mean free path

\lambda

100\,\mathrm{nm}

Collection semi-angle

\beta

8\,\mathrm{mrad}

Energy response · FWHM

\Delta E

3\,\mathrm{eV}

Detector dispersion

d_E

1\,\mathrm{eV/pixel}

Exposure per spectrum window

\tau

1\,\mathrm{s}

Probe position

x

0\,\mathrm{nm}

Synthetic edge shift

\delta E

0\,\mathrm{eV}

Predict how thinning changes repeated losses. Obtain relative thickness at most 0.35, energy response at most 1.5 eV, dispersion at most 0.5 eV per pixel and expected edge SNR at least 5 while probing the C-rich region. Then check the target.

Fixed 200 kV beam and 0.004 pA probe. The assumed low-loss mean free path is adjustable, not a material calibration. Independent 16 eV losses have 3 eV standard deviation and Poisson order statistics. Low-loss collection is assumed energy independent; the displayed log ratio uses actual counts within −10 to 130 eV and a finite zero-loss window, so broadening, noise and missing tails bias it. A separate rare-core-event approximation convolves a synthetic C-K edge near 284 eV with the low-loss distribution; its yields, near-edge feature and shift do not identify a real chemical state. Core acceptance uses an assumed dipole angular kernel at negligible convergence. Net core counts subtract the known model background; they are not a fitted elemental analysis. Gaussian instrument width is FWHM. Noise is seeded Poisson below 40 counts and a Gaussian approximation above. The magnetic sector has schematic dispersion of 0.1 scene unit per eV; no field or aberration solver is implied.

Start with the essentials

Focus question: What did the transmitted electrons lose?
One-sentence intuition: Zero loss does not mean no elastic scattering. Repeated low-loss events shift and broaden core-edge information, while detector dispersion and energy resolution describe different limits.

Core mathematical model

Repeated loss orders

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

Independent inelastic events give a zero-loss fraction and a plural-loss population. 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. The finite spectral window and finite zero-loss integration can bias the result, especially with broad response or thick specimens. Absolute thickness also needs a calibrated mean free path.

The instrument broadens, pixels integrate

\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 differ: narrowing detector bins does not undo an already broad energy response. Every bin integrates probability before 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})

This rare-core-event teaching approximation redistributes an edge toward higher loss. The synthetic onset and feature can be shifted, but no actual oxidation state is inferred. Known model-background subtraction gives an optimistic SNR reference.

Common difficulties

An edge shift is not a diagnosis

Typical misconceptionAny shifted edge names a unique oxidation state.

Better mental modelThese are invented near-edge shapes. Chemical identification requires references, calibration and a suitable scattering model.

Run the experiment

01
Thin, then thicken
Compare Thin · resolved with Plural scattering. Watch the low-loss orders and compare assumed thickness with the recorded-window estimate.
What to observe: Multiple losses grow and redistribute the core signal. A finite low-loss window can lose part of the tail.
02
Separate sampling from response
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 C-rich region, then restore it. Try the synthetic shift preset.
What to observe: The core edge weakens outside the inclusion. The low-loss mean stays fixed because this toy specimen shares the same mean free path.
04
Complete the target
Achieve the stated thinness, sampling, response and expected SNR requirements, then check.
What to observe: Known-background SNR is an optimistic reference. Real fitting, detector response and material cross sections need additional evidence.

Core mathematical model

Repeated loss orders

Relative thickness from counts

The instrument broadens, pixels integrate

A core event can be followed by low losses

Common difficulties

An edge shift is not a diagnosis

Run the experiment

Thin, then thicken

Separate sampling from response

Locate the signal

Complete the target