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Sandbox Physics

E35 · Energy-filtered imaging

EFTEM: Energy Windows & Elemental Maps

Slide an energy-selecting slit across the transmitted spectrum. Acquire two pre-edge images and one post-edge image, fit a power-law background per pixel, and compare the signed net map with a jump ratio and a separately acquired relative-thickness map. Recover a drifting acquisition without replacing it with specimen truth.

Interactive modelEFTEM: Energy Windows & Elemental Maps
Net map / blurred known amount correlation—\text{—}
Valid fitted overlap0 %0\,\mathrm{\%}
Assumed spatial blur · standard deviation0 nm0\,\mathrm{nm}
Five-image protocol dose—\text{—}
Background window geometry—\text{—}
Processed output—\text{—}
Experiment target—\text{—}

Physics tutorial

A filtered image is a measurement, not an element label

BackgroundAn energy filter passes a range of transmitted-electron losses while an image relay preserves specimen position. A post-edge image contains both core signal and a background.

Why it mattersThickness and drift can create bright regions even without a greater elemental amount. A defensible map needs acquired background windows and valid processing.

Start with the essentials

Focus question
Does a bright energy-filtered pixel mean more carbon?
One-sentence intuition
The three-window method fits and subtracts background at every pixel. Its residual depends on noise, window geometry, registration and the model of that background.

Core mathematical model

The slit integrates a spectrum at each pixel

Ck(r)=Ninc∫Ek−ΔE/2Ek+ΔE/2S(r,E) dEC_k(\mathbf r)=N_{\mathrm{inc}}\int_{E_k-\Delta E/2}^{E_k+\Delta E/2}S(\mathbf r,E)\,\mathrm dE

Counts are generated from this integral after the disclosed spatial blur and frame displacement. The live slit preview is a separate exposure from the three-image mapping protocol.

Fit integrated windows, not their centers

B(E)=A(E/284 eV)−rWk(r)=∫Ek−ΔE/2Ek+ΔE/2(E/284 eV)−r dEC1/C2=W1(r)/W2(r)\begin{aligned}B(E)&=A(E/284\,\mathrm{eV})^{-r}\\W_k(r)&=\int_{E_k-\Delta E/2}^{E_k+\Delta E/2}(E/284\,\mathrm{eV})^{-r}\,\mathrm dE\\C_1/C_2&=W_1(r)/W_2(r)\end{aligned}

The per-pixel fit solves for exponent and amplitude from actual pre-edge counts. Fits outside the assumed nonnegative exponent range up to eight are marked invalid. Edge leakage breaks the background assumption.

Subtract background; keep signed residuals

Cnet=Cpost−AWpost(r),Rjump=CpostC2C_{\mathrm{net}}=C_{\mathrm{post}}-A W_{\mathrm{post}}(r),\qquad R_{\mathrm{jump}}=\frac{C_{\mathrm{post}}}{C_2}

The net signal and the jump ratio have different meanings. Neither automatically yields atomic concentration: cross sections, thickness and plural scattering matter. Negative residuals are retained rather than clipped to create a cleaner image.

A separate low-loss pair estimates thickness

μapp(r)=ln⁡C0(r)+Closs(r)C0(r)\mu_{\mathrm{app}}(\mathbf r)=\ln\frac{C_0(\mathbf r)+C_{\mathrm{loss}}(\mathbf r)}{C_0(\mathbf r)}

Zero-loss and all-loss counts have the same assumed collection efficiency. Spatial blur and counting noise prevent exact pixelwise recovery. This relative-thickness image is separate from the carbon map.

Common difficulties

A jump ratio is not concentration

Typical misconceptionBrighter ratio pixels always contain a greater atomic fraction.

Better mental modelThe denominator includes background and thickness. A cross-section-calibrated quantitative analysis is outside this model.

Run the experiment

  1. 01

    Move the slit

    Compare losses near zero, 16 eV and 300 eV. Watch which representative paths pass the slit and how the detector changes.

    What to observe: The detector always displays the selected measured-energy window. The three-window mapping protocol uses its own fixed post window.
  2. 02

    Break a background assumption

    Choose Pre-edge contamination, then restore a safe second window. Widen the slit and inspect both boundaries.

    What to observe: The diagnostic rejects windows that overlap or include the carbon edge. A plausible-looking map is insufficient.
  3. 03

    Recover the acquired frames

    Choose Uncorrected drift, then enable known-frame alignment. Compare the raw frames, net profile, correlation and coverage.

    What to observe: Raw images remain unchanged. The processor resamples them and masks unsupported edges; counting noise remains.
  4. 04

    Compare different outputs and finish

    Try the jump-ratio and relative-thickness presets, then obtain a valid signed map with correlation at least 0.95 and coverage at least 75 percent.

    What to observe: Increasing exposure improves counting precision but increases protocol dose. The thickness and ratio outputs do not replace elemental background subtraction.