E01 · Electron microscopy / foundations

Electron Wavelength & Voltage

Set the accelerator voltage and compare classical and relativistic wavelengths. Watch the foil scattering profile and the maximum energy transferable to a carbon atom. Find the shortest wavelength below the selected knock-on threshold while keeping at least 40 percent unscattered electrons.

Selected wavelength $0\,\mathrm{pm}$

Classical wavelength $0\,\mathrm{pm}$

Speed as fraction of light speed $0$

Ideal diffraction scale $0\,\mathrm{nm}$

Unscattered fraction (illustrative) $0\,\mathrm{\%}$

Scattering width (illustrative) $0\,\mathrm{mrad}$

Maximum transfer to carbon $0\,\mathrm{eV}$

Knock-on threshold comparison $\text{—}$

Experiment task $\text{—}$

Exact free-electron kinematics; carbon-12 maximum-transfer approximation. Attenuation and angular profiles are illustrative scaling laws, without measured cross sections or Monte Carlo transport. The diffraction readout is an ideal circular-aperture scale. Being below a knock-on threshold does not exclude radiolysis, charging or heating.

Start with the essentials

Focus question: Is higher accelerating voltage always better?
One-sentence intuition: Shorter wavelengths and sample survival are separate constraints. A threshold can exclude knock-on events without excluding other damage mechanisms.

Core mathematical model

Relativistic wavelength

\lambda=\frac{h}{\sqrt{2m_e eV\left(1+\frac{eV}{2m_e c^2}\right)}}

Momentum, rather than the classical speed estimate, sets the matter-wave wavelength.

Ideal diffraction scale

d_{\mathrm{diff}}=0.61\lambda/\alpha

This does not include source size, aberrations, dose or detector sampling.

Heavy-nucleus energy transfer

T_{\max}\simeq\frac{2K(K+2m_ec^2)}{Mc^2},\quad K=eV

Carbon-12 is fixed. An adjustable threshold is an assumption, not a universal carbon material property.

Illustrative attenuation

P_0=\exp(-t/\ell),\quad \ell=80\,\mathrm{nm}\left(\frac{V}{100\,\mathrm{kV}}\right)^{0.65}

The chosen scaling creates a teaching comparison; it is not a calibrated cross-section model.

Common difficulties

Keep the model boundary explicit

Typical misconceptionA beam below the knock-on threshold is harmless.

Better mental modelRadiolysis, heating, contamination and charging require additional material and dose models.

Run the experiment

01
Predict
Predict how 300 kV changes wavelength and energy transfer relative to 80 kV, then compare the presets.
What to observe: Wavelength falls while maximum transfer rises.
02
Find the safe edge
Keep transfer below the assumed threshold but at least 85 percent of it; keep the unscattered fraction above 40 percent. Press Check target.
What to observe: The task combines a resolution objective with two specimen constraints.
03
Challenge the approximation
Turn off the relativistic model at high voltage, then increase foil thickness.
What to observe: The classical wavelength overestimates the relativistic value; thickness reduces the unscattered beam.