A projector lens system does not image reciprocal space perfectly. The best-documented of its errors is elliptical distortion: the magnification differs along different radial directions, so a circle is recorded as an ellipse. It is first order in radius, meaning it is the same percentage at every radius, and it is a linear map, so it carries a lattice to another lattice. That last property is what makes it so dangerous. Your spot pattern still looks like a perfectly good lattice. It is simply the wrong one.
Measured magnitudes from three independent groups: 1.6 to 1.9% across camera lengths on one instrument (Capitani et al. 2006); a 2% installation specification with 1.6% as installed, falling to 0.3% after stigmator optimisation (Mitchell 2022); and a refined 0.38 to 0.39% on a well-characterised machine (Brázda et al. 2022). At 1% and a reflection 500 px from the centre, that is a 5 px radial displacement, two orders of magnitude above the precision a modern peak finder reaches.
A single spot pattern cannot measure it. A two-dimensional net gives you two shape observables, a length ratio and an included angle, while the distortion adds two unknowns, an amplitude and an axis. The problem is underdetermined, and Brázda et al. reach the same conclusion from the other direction: elliptical distortion is the one distortion that cannot be separated from the lattice parameters without external information. A ring pattern supplies exactly that external information, because a ring has a known shape. It should be a circle. Whatever it actually is tells you the map the optics applied.
Each ring is measured as a radius against azimuth, in wedges rather than along single rays (a polycrystalline ring is made of discrete spots, so a single ray usually lands between them), and the result is fitted with the first few Fourier terms. That is the same decomposition the distortion literature uses:
r(θ) = A₀ + A₁cosθ + B₁sinθ + A₂cos2θ + B₂sin2θ + A₃cos3θ + B₃sin3θ
The radius itself is measured twice per ring. The first pass uses a window wide enough to be sure of catching the ring, which biases the measurement toward the window centre and so understates the ellipse. The second pass re-centres the window on the ellipse the first pass found and narrows it to the ring's own width, at which point the window has no preferred radius and the bias is gone.
Elliptical distortion is first order in radius, so its amplitude is the same on every ring. Barrel and pincushion distortion is third order, so it grows as r². One ring cannot tell them apart; three or four can, and this page reports both. It also uses the agreement between rings as a check: a ring whose ε disagrees with the others is not measuring the ellipse, and is rejected and flagged rather than averaged in.
This causes real confusion when comparing papers, so it is worth stating plainly. For an ellipse with semi-axes a and b:
That conversion reconciles two results that look wildly different. Czigány and Kis (2022) report raw eccentricities between 0 and 0.17 across instruments, and 0.13 ± 0.01 on their own; Brázda et al. refine an amplitude of 0.38 to 0.39%. An eccentricity of 0.13 is an amplitude of 0.43%. The two are saying the same thing.
Once the rings are fitted, R × d = λL for every one of them, so a known calibrant also gives the camera constant. The rings are not assumed to be the first n of the calibrant's list, because the innermost may be behind the beamstop and a weak one may be missed: every plausible assignment is tried, each predicts where all the other rings should fall, and the one that actually places them wins.
The acceptance test is the standard suppliers': the λL values computed from different rings should agree within 1%, and their mean is used thereafter (Ted Pella technical note for standard 619). A systematic drift of λL with ring radius is not scatter, it is barrel or pincushion distortion, and it is fitted and reported separately rather than averaged away. Note what this does and does not fix: a pure camera-length error preserves every ratio and every angle, so it does not affect a zone axis at all. It wrecks phase identification, which depends on absolute d-spacings.
For what is achievable: Czigány and Kis report 0.1% absolute accuracy in lattice spacing and 0.02 to 0.03% reproducibility of the camera length between sessions, and note that 0.1% is comparable to routine laboratory X-ray diffraction.
The criteria, in Czigány and Kis's own terms, are that the specimen be polycrystalline with narrow rings, free of texture, of known structure and lattice parameter, and with non-overlapping rings in the angular range you use. Gold is the usual internal standard: inert, beam-hard, strong well-separated rings, and the best-agreed lattice parameter of any metal offered here. Evaporated aluminium is the common commercial product, sold on its very fine as-evaporated grain size, though its rings are weaker and it carries a native oxide. Silicon has by far the best-known lattice parameter, NIST-certified and traceable to the metre, but it is not a fine-grained evaporated film.
Two things are deliberately not offered. Thallous chloride is a classic standard, but its published lattice parameter is unresolved between 3.834 and 3.8416 Å, a 0.2% conflict that would become a 0.2% camera-constant error for everyone who used it; it is also an insulating halide, and so sits in the radiolysis-dominant damage class, and thin films of it are reported to adopt different structures depending on the substrate. Amorphous carbon is a perfectly good distortion target, because a halo is a closed isotropic ring whose ellipticity can be fitted without knowing its radius, but it has no defensible nominal d value: halo positions shift by more than 15% with preparation and heat treatment. Use it for ε, never for λL.
More TEM tools on this site: the SAED zone-axis indexer · how to index a pattern · the Ewald sphere · about the author
@misc{tripathy2026diffractiondistortioncalibrator,
author = {Tripathy, Manisha},
title = {Diffraction Distortion Calibrator},
year = {2026},
howpublished = {\url{https://untetheredatom.com/tem/diffraction-distortion-calibrator}},
note = {Interactive teaching resource}
}