Rings in your FFT? The lattice is gossiping about your sample prep.
High-resolution TEM shows you rows of dots at atomic spacings, and the FFT of that image looks just like a diffraction pattern. Both facts are seductive and both are traps: the dots are not simply atoms, and the FFT is not simply diffraction. This guide, built around the questions everyone actually asks, lets you form the image yourself, break it yourself, and measure from it correctly. Example material throughout: Cu (d₁₁₁ = 0.209 nm, d₂₀₀ = 0.181 nm).
This panel builds an HRTEM image of a Cu foil viewed along [011] the way the microscope does: take the projected potential of the atomic columns, pass it through the objective lens's contrast transfer function (CTF), and see what arrives. Drag the defocus. The dots move, sharpen, blur, and (the crucial party trick) swap between black and white. Then switch on the column-position overlay and check whether the dots are even on the atoms.
A perfect lens would transfer every spacing faithfully. A real objective multiplies each spatial frequency u by sin χ(u), where χ = πλΔf·u² + ½πCsλ³u⁴: an oscillating function that passes some spacings positive, some negative (contrast reversed!), and some not at all. Scherzer defocus is the Δf that stretches the first broad negative band as wide as possible; the first zero after it defines the point resolution. (Sign convention: underfocus is negative here, so Scherzer sits near −66 nm on this 200 kV setup; other texts and software define Δf with the opposite sign.) Coherence envelopes damp everything beyond, down to the information limit.
The FFT of a lattice image concentrates each set of fringes into a pair of sharp spots: distance from centre = 1/d, direction ⊥ to the fringes. That makes it a superb local d-spacing and orientation tool. Drag the region-of-interest box around the image below (over one grain, the other, the boundary, the amorphous edge) and watch its FFT. Then click any spot in the FFT to measure it.
Mask a few spots in the FFT, inverse-transform, and a noisy image becomes a clean set of fringes. Used carefully this is legitimate (isolating one grain's fringes, making strain fields visible). Used carelessly it manufactures lattice where none exists: a mask passes noise too, and noise through a periodic mask looks periodic. Try all three masks below on the same noisy image; the third one should worry you.
They share geometry; that's why FFT spot positions index like SAED spots. But they are physically different objects, and the differences matter exactly when you're tempted to over-interpret:
| Property | SAED (real diffraction) | FFT of an HRTEM image |
|---|---|---|
| What it transforms | The actual specimen exit wave: physics does the Fourier transform | The recorded intensity image, after the lens already scrambled phases |
| Intensities mean… | |structure factor|² × dynamical effects: real crystallographic information | (fringe contrast)² × CTF²: lens settings in, crystallography mostly out |
| Resolution limit | Bragg angles out to very high g (Å⁻¹ and beyond) | Hard cut at the information limit AND at Nyquist (2 px per fringe) |
| Region selected | SA aperture: ≳ 100 nm circle (or the beam in μ/nano-diffraction) | Any ROI you like, down to a few nm: its unbeatable advantage |
| Weak/forbidden spots | Present if dynamically excited | Only if the corresponding fringes beat the image noise floor |
| Good for | Phase ID, orientation, true intensities, unknown structures | Local d-spacings & orientations, defect/strain analysis, quick lattice checks |
@misc{tripathy2026hrtemffts,
author = {Tripathy, Manisha},
title = {HRTEM \& FFTs: Lattice Fringes, Honestly},
year = {2026},
howpublished = {\url{https://untetheredatom.com/tem/hrtem-fft-guide}},
note = {Interactive teaching resource}
}