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 broad, blurred dark line 10 to 15 nm wide. Tilt the crystal so your reflection is weakly excited (the g–3g condition), and the same dislocation becomes a sharp bright line about 1.5 nm wide on a nearly black background. This guide is built around the questions people ask about how and why that works, with live simulations. Example material throughout: Cu, 220 reflection, 100 kV (the case the method was invented on).

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 the Cu 220 reflection at 100 kV unless stated: ξ₂₂₀ ≈ 44 nm (the SAED pattern simulator on this site, Peng 1996 scattering factors; the Hirsch et al. table gives a few per cent less). At 200 kV the extinction distance is about 1.3 times longer and, for the same g(3g) tilt, s_g is smaller (see the box in the next part).

Why are my dislocation images broad and blurred?

Diffraction contrast has a natural length scale: the extinction distance ξ_g, the depth over which intensity passes back and forth between the direct and diffracted beams. At the exact Bragg condition that exchange is slow (ξ₂₂₀ ≈ 44 nm in Cu at 100 kV), and a dislocation perturbs the image over about ξ_g/3, 10 to 15 nm. Tilt away from Bragg and the exchange speeds up: the effective extinction distance shrinks as ξ_eff = ξ_g/√(1+(s·ξ_g)²). A shorter depth period gives a sharper image. Drag s and watch.

Pendellösung intensity vs depth in a perfect crystal

What it costs & buys vs excitation error s

Excitation error s 0.00 nm⁻¹

43.9 nm

ξ_eff: depth period

14.6 nm

≈ dislocation image width (ξ_eff/3)

100%

background intensity (rel. to s=0)

What does g–3g mean, and how do I set it up?

"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 s_g), 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.

Setup step

Ewald sphere systematic row, side view

Diffraction pattern with Kikuchi lines

Fine crystal tilt 0.0 · s₉ = 0

The rule of thumb behind the recipe: a good weak-beam image needs s_g ≳ 0.2 nm⁻¹ (so ξ_eff ≲ 5 nm in Cu). With the reflection ng at Bragg, the excitation error of g is s_g = (n−1)λ|g|²/2. For Cu 220 at 100 kV, the case Cockayne, Ray and Whelan used in 1969, putting 3g at Bragg gives s_g = λ|g|² ≈ 0.23 nm⁻¹, which is why g–3g became the standard. At 200 kV the same tilt gives only 0.15 nm⁻¹ for Cu 220 (0.08 nm⁻¹ for Cu 200), so at 200 kV and above use g–4g or g–5g: Cu 220 with 5g at Bragg gives 0.31 nm⁻¹ at 200 kV. Materials with bigger unit cells (smaller |g|) need a higher n for the same reason. 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 idea of weak beam. Move s and watch the broad strong-beam image narrow to 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

Excitation error s 0.00 nm⁻¹

Foil thickness 80 nm

g·b 2

Dislocation depth 40 nm

–

image FWHM

–

peak offset from core

–

peak / background contrast

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

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. (For edge dislocations a weak residual image 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 splits into two Shockley partials (b = ⅙⟨112⟩) with a ribbon of stacking fault between them. Their separation d is set by a balance: elastic repulsion pushes them apart, the fault energy γ pulls them together, so measuring d measures γ. The difficulty is that in Cu, d is only a few nm. A strong-beam image (width 10 to 15 nm) shows one blurred line; 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)

Stacking-fault energy γ 45 mJ/m²

Dislocation character edge (90°)

–

partial separation d

–

resolved in WBDF?

–

material at this γ (roughly)

References

Show the 7 references

D. J. H. Cockayne, I. L. F. Ray and M. J. Whelan, Investigations of dislocation strain fields using weak beams, Phil. Mag. 20, 1265-1270 (1969): the weak-beam method and the g(3g) condition.
D. J. H. Cockayne, The principles and practice of the weak-beam method of electron microscopy, J. Microsc. 98, 116-134 (1973): the s_g ≥ 0.2 nm^-1 rule and the practical recipe.
D. J. H. Cockayne, M. L. Jenkins and I. L. F. Ray, The measurement of stacking-fault energies of pure face-centred cubic metals, Phil. Mag. 24, 1383-1392 (1971): partial dislocation separations in Cu and the stacking-fault energy.
A. Howie and M. J. Whelan, Diffraction contrast of electron microscope images of crystal lattice defects. II. The development of a dynamical theory, Proc. R. Soc. A 263, 217-237 (1961); and A. Howie and M. J. Whelan, Diffraction contrast of electron microscope images of crystal lattice defects. III. Results and experimental confirmation of the dynamical theory of dislocation image contrast, Proc. R. Soc. A 267, 206-230 (1962): the equations the live simulation integrates, the g·b = 0 rule and the g·(b×u) residual.
P. B. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley and M. J. Whelan, Electron Microscopy of Thin Crystals, 2nd ed., Krieger (1977): dislocation image contrast, the column approximation and the extinction distance tables.
D. B. Williams and C. B. Carter, Transmission Electron Microscopy: A Textbook for Materials Science, 2nd ed., Springer (2009): the chapters on imaging strain fields and on weak-beam dark-field microscopy.
J. W. Edington, Interpretation of Transmission Electron Micrographs (Practical Electron Microscopy in Materials Science, Monograph 3), Macmillan (1975): setting up two-beam and weak-beam conditions at the microscope.

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 16 October 2026.