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Sandbox Physics

E06 · Electron microscopy / specimen physics

Electron Scattering & Mean Free Paths

Change energy, thickness, atomic number and collection angle. Watch scattering broaden the angular screen and split the energy-loss spectrum into successive orders. Find a useful thickness with enough collected zero-loss electrons.

Interactive modelElectron Scattering & Mean Free Paths
Elastic mean free path (proxy)0 nm0\,\mathrm{nm}
Inelastic mean free path (proxy)0 nm0\,\mathrm{nm}
Zero-loss fraction0 %0\,\mathrm{\%}
No-event fraction0 %0\,\mathrm{\%}
Plural inelastic fraction0 %0\,\mathrm{\%}
Gaussian angular width0 mrad0\,\mathrm{mrad}
Collected zero-loss fraction0 %0\,\mathrm{\%}
Mean energy loss0 eV0\,\mathrm{eV}
Probability inside displayed spectrum0 %0\,\mathrm{\%}
Experiment task—\text{—}

Physics tutorial

How thin must a specimen be to avoid plural scattering?

BackgroundChange energy, thickness, atomic number and collection angle. Watch scattering broaden the angular screen and split the energy-loss spectrum into successive orders. Find a useful thickness with enough collected zero-loss electrons.

Why it mattersSeparate specimen physics from detector appearance before interpreting an electron image.

Start with the essentials

Focus question
How thin must a specimen be to avoid plural scattering?
One-sentence intuition
Zero-loss does not mean unscattered: elastic deflections change direction without contributing to the loss spectrum.

Core mathematical model

Poisson loss orders

Pn=μne−μn!,μ=t/ℓiP_n=\frac{\mu^n e^{-\mu}}{n!},\quad \mu=t/\ell_i

A fixed excitation quantum isolates the role of specimen thickness; real materials have continuous and discrete loss channels.

No-event and zero-loss probabilities

PZL=e−t/ℓi,Pnone=e−t(1/ℓi+1/ℓe)P_{\mathrm{ZL}}=e^{-t/\ell_i},\quad P_{\mathrm{none}}=e^{-t(1/\ell_i+1/\ell_e)}

Elastic scattering removes electrons from the direct beam without removing them from the zero-loss energy channel.

Plural scattering

Pn≥2=1−e−μ(1+μ)P_{n\ge2}=1-e^{-\mu}(1+\mu)

The plotted zero-loss probability includes no angular aperture; the collected readout does.

Common difficulties

Model boundary

Typical misconceptionA schematic image is a calibrated material prediction.

Better mental modelIndependent Poisson inelastic events, a fixed 16 eV loss quantum and a Gaussian small-angle elastic envelope. Mean-free-path scaling is illustrative, without cross-section tables. The aligned-crystal switch only reduces an assumed elastic rate; it does not solve dynamical diffraction. The fixed spectrum window can miss high loss orders.

Run the experiment

  1. 01

    Predict

    Predict how increasing thickness changes the zero-loss peak and higher loss orders; compare the thin and thick presets.

    What to observe: Zero-loss does not mean unscattered: elastic deflections change direction without contributing to the loss spectrum.
  2. 02

    Tune and check

    Keep thickness at least 20 nm, zero-loss collection at least 60 percent, and plural inelastic scattering below 10 percent.

    What to observe: The task checks quantitative readouts rather than visual brightness.
  3. 03

    Explain the limit

    Shrink collection angle while keeping thickness fixed. Explain why less collected zero-loss intensity does not imply a shorter inelastic mean free path.

    What to observe: Independent Poisson inelastic events, a fixed 16 eV loss quantum and a Gaussian small-angle elastic envelope. Mean-free-path scaling is illustrative, without cross-section tables. The aligned-crystal switch only reduces an assumed elastic rate; it does not solve dynamical diffraction. The fixed spectrum window can miss high loss orders.