untetheredatom
Interactive TEM guide · No. 5

Why are my dislocation images too wide?

The dislocations are not wide, the imaging condition is (weak beam fixes both).

In a strong two-beam image a dislocation is a fat, fuzzy worm ~10 nm wide. Tilt the crystal so your reflection is weakly excited (the famous g–3g condition), and the same dislocation collapses into a crisp bright line ~1.5 nm wide on a nearly black background. This guide is built around the questions everyone actually asks about how and why that works, with live simulations to play with. Example material throughout: Cu.

🔬
Scope: applies to any TEM that can do dark-field imaging: all you need is a tilt stage, an objective aperture, and dark-field beam tilt. Numbers are for Cu at 200 kV unless stated (ξ200 ≈ 27 nm).

Why are my dislocations fat, blurry worms?

Diffraction contrast has a natural length scale: the extinction distance ξg, the depth over which intensity sloshes between the direct and diffracted beams. At the exact Bragg condition that slosh is slow (ξ200 ≈ 27 nm in Cu at 200 kV), and a dislocation perturbs the image over ~ξg/3 ≈ 9–13 nm. Tilt away from Bragg and the slosh speeds up: the effective extinction distance shrinks as ξeff = ξg/√(1+(s·ξg)²). Sharper slosh → sharper image. Drag s and watch.

Pendellösung intensity vs depth in a perfect crystal

What it costs & buys vs excitation error s

27.0 nm
ξ_eff: depth period
9.0 nm
≈ dislocation image width (ξ_eff/3)
100%
background intensity (rel. to s=0)

What does g–3g actually mean, and how do I dial it in?

"g–3g" is a tilt recipe, read as: form the image with beam g, while tilting so that 3g sits exactly on the Bragg condition. That parks g far off Bragg (large sg), weakly excited, hence "weak beam", but in a reproducible, quotable way. Walk the steps below; the Ewald sphere, the systematic row, and the Kikuchi lines all move together.

Ewald sphere systematic row, side view

Diffraction pattern with Kikuchi lines

The rule of thumb behind the recipe: a good weak-beam image needs sg ≳ 0.2 nm⁻¹ (so ξeff ≲ 5 nm in Cu). For Cu 200 at 200 kV, putting 3g at Bragg gives sg ≈ 0.24 nm⁻¹, which is why g–3g became the standard. In materials with bigger unit cells you may need g–4g or g–5g to reach the same s; the condition is a means, s is the goal.

What will I actually see? A live two-beam simulation

This panel integrates the Howie–Whelan equations, the real dynamical two-beam theory, column by column through a foil containing one screw dislocation (line running into the screen). Far from the core the crystal is far off Bragg and stays dark; right beside the core the planes are locally bent back into the Bragg condition, and that narrow column lights up. That is the whole trick of weak beam. Play with s and watch the strong-beam worm collapse into a sharp line, and notice the bright line sits slightly beside the true core.

Dark-field intensity profile across the dislocation

Simulated DF image line direction vertical · ▲ marks true core

image FWHM
peak offset from core
peak / background contrast

When do dislocations vanish? The g·b = 0 game

A dislocation only distorts planes that its displacement field actually bends: if the imaging reflection g is perpendicular to the Burgers vector b (so g·b = 0), those planes stay flat and the dislocation disappears. Flip through reflections and identify each segment's b: exactly the two-tilt experiment you'd run at the microscope. (Fine print: for edge dislocations a weak residual ghost survives unless g·(b×u) is also small.)

Choose imaging reflection g (Cu foil, [011] zone)

WBDF image three segments, three different b

b₁ = ½[10̄1]b₂ = ½[110]b₃ = ½[01̄1]

Can I really see partial dislocations? (Measuring Cu's stacking-fault energy)

In fcc Cu every perfect dislocation is secretly two Shockley partials (b = ⅙⟨112⟩) bracketing a ribbon of stacking fault. Their separation d is set by a tug-of-war: elastic repulsion pushes them apart, the fault energy γ pulls them together, so measuring d measures γ. The catch: in Cu, d is only a few nm. A strong-beam image (width ~10 nm) shows one blurred worm; weak beam (width ~1.5 nm) resolves the pair. This is precisely the experiment Cockayne, Ray & Whelan invented the technique for in 1969.

Strong beam, s ≈ 0 the same dislocation

Weak beam, g–3g partials resolved (when possible)

partial separation d
resolved in WBDF?
material at this γ (roughly)

Why is my image nearly black? (practical checklist & FAQ)

Weak beam is dim by design: you deliberately threw away almost all the diffracted intensity to buy sharpness. Everything practical about the technique follows from that trade.

Cite this page: Tripathy, Manisha. “Weak-Beam Dark Field: Seeing Dislocations Sharp.” untethered atom, 2026, https://untetheredatom.com/tem/weak-beam-dark-field-guide.
BibTeX
@misc{tripathy2026weakbeamdarkfield,
  author = {Tripathy, Manisha},
  title  = {Weak-Beam Dark Field: Seeing Dislocations Sharp},
  year   = {2026},
  howpublished = {\url{https://untetheredatom.com/tem/weak-beam-dark-field-guide}},
  note   = {Interactive teaching resource}
}
Last updated 13 August 2026.