Interactive TEM guide · No. 6

Why does my HRTEM FFT show rings or Moiré?

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).

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Scope: any TEM doing phase-contrast imaging. Defaults are 200 kV (λ = 2.51 pm); the C_s toggle switches between an uncorrected objective (C_s = 1.2 mm) and an image-corrected one (C_s ≈ 5 µm). Simulations use the weak-phase-object approximation: the thin-foil limit where the theory is cleanest.

Are those dots atoms? Form the image and see

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.

Simulated HRTEM image Cu [011], ~3.2 nm field

CTF at this setting sin χ(u) · envelope: Cu reflections marked

Defocus Δf −66 nm (Scherzer)

Aberration-correctedCₛ 1.2 mm → 5 µm (and the Scherzer game changes completely) Show true column positionsteal dots = where the atoms actually are

The honest summary: an HRTEM image is an interference pattern whose relationship to the atoms is set by defocus, thickness, and the lens, not a photograph. Dots can sit on columns, between columns, or reverse contrast entirely, and in thicker foils (beyond this weak-phase simulation) they cycle with thickness too. Never say "the bright dots are the atoms" without a simulation to back it up.

The CTF: the lens as a scrambler of spatial frequencies

A perfect lens would transfer every spacing faithfully. A real objective multiplies each spatial frequency u by sin χ(u), where χ = πλΔf·u² + ½πC_sλ³u⁴: 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.

CTF explorer sin χ with partial-coherence envelope

Defocus Δf −66 nm

Spherical aberration Cₛ 1.2 mm

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Scherzer defocus

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point resolution (first zero)

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information limit (envelope)

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Cu {111} transferred as…

Measuring d-spacings with the FFT: the right way

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.

HRTEM-like image drag the ROI box

FFT of the ROI log power · click a spot to measure

Window the ROI edges (Hann)uncheck to see the bright cross artifact that puzzles everyone

Detector sampling 0.05 nm/px

coarser pixels shrink the Nyquist circle; spacings finer than 2 px/fringe are gone

Specimen drift during the exposure0.35 nm of travel at 20°, so the image is smeared along one direction Projector or scan distortiona 4% shear at 30°, the kind a badly aligned projector or a drifting scan puts on every image Saturated detectorthe top of the intensity range clipped flat, as when the exposure is pushed for contrast

Pixel-size calibration error 0%

the image and the FFT spots do not change: only your scale does. The dashed ring shows where your scale puts Cu {111}

click a spot

measured d-spacing

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fringe normal angle

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closest Cu match

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what it should be

Calibration reality check: an FFT measurement is only as good as your pixel-size calibration and ROI statistics: one spot a few pixels from centre carries huge fractional error. Best practice: use the largest clean ROI you can, measure the pair separation (2/d) and halve it, and sanity-check against a known spacing in the same image (here: Cu {111} = 0.209 nm).

Four ways the FFT lies, and what each looks like. The toggles above put each one on the same image, so you can learn the signature rather than the excuse.

Drift smears the image along one direction, and a smear is a convolution, so the FFT gets multiplied by a sinc along that same direction. Spots perpendicular to the drift survive; spots along it fade and can disappear entirely. A one-sided FFT with a healthy-looking image is almost always drift, not crystallography.
Distortion is an affine map on the image, so it is an affine map on the FFT too. Every d-spacing you measure is wrong by a direction-dependent amount, and the angle between two spot pairs stops matching the crystal. That angle is the check: for the Cu {111} pairs here it must be 70.5°, and it is not a free parameter.
Saturation flattens the peaks of the fringes, which is a nonlinearity, and a nonlinearity makes harmonics. Second-order spots appear at 2g where the crystal has none. Reported as a superlattice, this is a real and recurring error in the literature.
Calibration error shows nothing at all. The image is fine, the FFT is fine, the spots are sharp, and every spacing is wrong by the same percentage. The only defence is a known spacing in the same image, which is why the amorphous region and a reference lattice are worth keeping in frame.

The fifth case, an ROI whose edges cut the lattice, is the Hann toggle above: that one is an artifact of how you cropped, not of how you imaged.

Fourier filtering: powerful, and quietly dangerous

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.

Input noisy Cu lattice image + its FFT

Inverse FFT of masked spectrum result

Pass both {111} spot pairslegitimate: cleaned lattice image Pass one pair onlyone fringe system: used for Burgers-circuit / GPA-style analysis Pass two empty noise regionsno real periodicity inside the mask at all

Is the FFT the same as a diffraction pattern?

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

Rule of thumb: index and measure with the FFT, but identify phases with real diffraction (SAED/CBED/nanodiffraction): an FFT spot at 0.21 nm tells you a 0.21 nm periodicity survived your lens and camera, not that it belongs to Cu {111}.