TEM · Aberration correction

Aberration correction in the TEM and STEM: what it fixes and what it leaves

Start with the ray picture: pick a hexapole corrector and drag it until every ray meets at one point. Then see what still limits the image.

1 Rays and the corrector 2 HRTEM transfer and the information limit 3 STEM probe size 4 Residual aberrations

What does spherical aberration do, and how does a corrector cancel it?

A round magnetic lens bends rays far from the axis too strongly, so they cross the axis before the focus. This is spherical aberration, and its size is the coefficient Cs (also written C3). Scherzer proved in 1936 that Cs of a round, static lens is always positive. A corrector uses non-round fields (hexapoles, or quadrupoles and octupoles) to add a negative Cs.

Left: the lens and corrector (not to scale). Right: the focus region to scale: each ray crosses the axis Csθ2 before the focus. The red bar is the disc of least confusion; the blue band is the diffraction spot 1.22λ/α.
Corrector
Total Cs
Focus spread Csα2
Disc of least confusion ½Csα3
Diffraction spot 1.22λ/α
Try it: at 200 kV, Cs 1.2 mm and α 10 mrad the disc of least confusion is 0.60 nm, twice the 0.31 nm diffraction spot. Press Hexapole and drag the corrector to −1.20 mm: the rays meet at one point and only the diffraction spot is left. Drag further to −2 mm: now the outer rays focus long.

Why does a corrected TEM still have a resolution limit?

In HRTEM the lens adds a phase χ(u) to each spatial frequency u. The image shows sinχ, the contrast transfer function (CTF). Two envelopes damp it: the temporal envelope from chromatic aberration Cc and the energy spread ΔE, and the spatial envelope from the beam convergence. With Cs near zero, sinχ no longer limits the image. The temporal envelope does: the information limit.

Left: sinχ (grey), the envelopes (dashed) and the damped CTF (blue). Dotted: point resolution; red: information limit. Ticks: Si reflections along [110], darker when they transfer strongly. Right: weak-phase HRTEM image of Si [110], 3 × 2 cells, same grey scale for every setting; circles mark the atom columns.
Microscope
Scherzer defocus for this Cs
Scherzer defocus
Point resolution
Information limit
Si (004), 0.136 nm: transfer
Try it: the uncorrected 200 kV FEG (Cs 1.2 mm, Cc 1.2 mm, ΔE 0.7 eV) gives Scherzer defocus −65.8 nm, point resolution 0.245 nm and information limit 0.139 nm: the dumbbells are one dark blob. Press "Cs corrected": the point resolution drops to 0.062 nm, but the information limit stays at 0.139 nm. Press "Corrected + monochromator": ΔE 0.3 eV moves it to 0.091 nm and the pairs split clearly.

How small can a STEM probe be, and what does the corrector change?

In STEM the lens focuses the beam to a probe. A larger convergence angle α gives a smaller diffraction spot, but Cs spoils the outer part of the cone. Without a corrector the best α is about 1.41(λ/Cs)1/4 (Kirkland), near 9.5 mrad at 200 kV. With Cs corrected, the fifth-order term C5 sets a new, larger optimum.

Top left: probe intensity across the probe (same current, so peak heights compare). Bottom left: probe size d50 (diameter holding half the current) against α, each point at its best defocus. Right: incoherent (HAADF-like) image of Si [110] dumbbells, columns 0.136 nm apart, and the profile across one pair.
Probe
Best α and Δf for this lens
Probe FWHM
d50 (with source)
Diffraction limit 0.61λ/α
Dumbbell: middle / peak
Try it: the corrected 25 mrad probe (Cs 1 µm) is 0.051 nm wide at half maximum, and the Si dumbbells split with the middle at 18% of the peak. Press "Uncorrected, 9.5 mrad": the probe is 0.14 nm wide and the pair is one blob. Then open α to 20 mrad on the uncorrected lens: the probe gets worse, not better.

Which aberrations are left after correction, and how small must they be?

A corrector removes Cs but can leave other terms: defocus C1, twofold astigmatism A1, axial coma B2, threefold astigmatism A2, and the next round terms C3 and C5. Each adds phase across the aperture. A common rule: keep the phase within ±π/4 of flat over the whole aperture. Names follow Haider and Krivanek (see the table under More detail).

Top: phase χ across the aperture (red +, blue −, black line at |χ| = π/4, dashed circle: the flat region); the probe it makes (intensity relative to a perfect lens); the Si [110] image with that probe. Bottom: Zemlin tableau (diffractograms of thin amorphous carbon at 18 mrad beam tilt in eight directions, 25 nm underfocus added; grey where the rings are finer than a pixel) and each term's phase at the aperture edge against π/4.
Start from
Flat to π/4 out to
Probe peak / perfect lens
Largest term at the edge
Dumbbell: middle / peak
Try it: from "Tuned corrector", raise A1 to 3 nm: the phase map turns into a saddle, the probe spreads into a four-armed cross and its peak drops to about 40% of perfect. Set it back and raise B2 to 100 nm: the probe turns lopsided, and in the tableau the rings on opposite sides of the centre stop matching. That mismatch is what coma-free alignment removes.

What to take away

Round lenses cannot fix themselvesCs of a round, static lens is always positive (Scherzer 1936). A corrector adds non-round fields that give negative Cs, so the total can be zero.
Then the information limit rulesWith Cs near zero, HRTEM transfers detail far past the old point resolution. The limit is then set by Cc, the energy spread and stability, not by Cs.
A corrected probe uses a bigger coneAt 200 kV the best uncorrected probe uses about 9.5 mrad and is about 0.14 nm wide. Corrected, 25 mrad gives about 0.05 nm and resolves Si dumbbells.
Corrected is not aberration-freeDefocus, astigmatism, coma, threefold astigmatism and C5 remain and drift. Each must stay within about π/4 of phase over the aperture, so the corrector is tuned again and again.
More detail: the formulas, the notation and where the models stop

The aberration function. For a beam at angle θ and azimuth φ from the axis, χ(θ,φ) = (2π/λ)[½C1θ2 + ½A1θ2cos2(φ−φA1) + B2θ3cos(φ−φB2) + ⅓A2θ3cos3(φ−φA2) + ¼C3θ4 + ⅙C5θ6]. With θ = λu this is χ(u) = πλΔf u2 + (π/2)Csλ3u4 + (π/3)C5λ5u6 for the round terms, the same form as the SAED pattern simulator. Negative Δf is underfocus.

Name here (Haider)KrivanekAberration±π/4 limit at α = 25 mrad, 200 kV
C1 (Δf)C1,0defocus1.0 nm
A1C1,2twofold astigmatism1.0 nm
B2C2,1 = 3B2axial coma20 nm
A2C2,3threefold astigmatism60 nm
C3 (Cs)C3,0spherical aberration3.2 µm
C5C5,0fifth-order spherical7.7 mm

The limit for a term of order n (phase grows as θn+1) falls as αn+1: doubling the aperture makes the C3 limit 16 times tighter. That is why a larger probe angle needs a better-tuned corrector.

Ray picture. A ray at angle θ crosses the axis Csθ2 before the Gaussian focus. The smallest spot, the disc of least confusion, has diameter ½Csα3 and sits ¾Csα2 before the focus. Rays ignore diffraction, so widget 1 adds the diffraction spot 1.22λ/α separately; widgets 2 to 4 use wave optics.

CTF and envelopes. CTF = sinχ·Et·Es. Temporal envelope Et = exp[−½(πλδu2)2] with focus spread δ = Cc(ΔE/E)(1+E/E0)/(1+E/2E0), E0 = 511 keV. Spatial envelope Es = exp[−(πα/λ)2(Csλ3u3 + Δfλu)2]. Scherzer defocus −1.2√(Csλ), point resolution 0.66(Csλ3)1/4, information limit where Et = e−2: d = (πλδ/2)1/2. For negative Cs the page uses +1.2√(|Cs|λ) (overfocus, bright atoms). The image is the linear weak-phase image of Si [110] (Peng 1996 scattering factors, B = 0.46 Å2, peak phase 0.3 rad): I = 1 + 2Σφgsinχ(g)E(g)cos(2πg·r). Real crystals more than a few nm thick scatter dynamically; the simulator's tableau handles that.

Probe. ψ(r) = ∫ exp(−iχ) J0(2πkr) 2πk dk over the aperture, normalised to unit current. d50 is the diameter holding half the current; the source size is added in quadrature to d50 and as a Gaussian blur in the image. The STEM images are incoherent sums of the probe intensity at point columns: no channelling, no thermal diffuse scattering, so relative column brightness is not quantitative (see quantitative HAADF). Widget 4 builds the probe by FFT on a periodic grid, without source blur.

Zemlin tableau. For beam tilt τ, the diffractogram of a thin amorphous film is sin2{[χ(τ+u) + χ(τ−u) − 2χ(τ)]/2}, damped here by a temporal envelope. For a tilt τ, coma B2 adds a defocus 4B2τ and an astigmatism 2B2τ that change sign with the tilt direction, and C3 adds about 2C3τ2 of defocus in every direction; it is zero when opposite tiles match (coma-free alignment, Zemlin et al. 1978). The π/4 test here is |χ| ≤ π/4 measured from the centre of the aperture, one of several published criteria (Uhlemann and Haider 1998 use the same quarter-wave idea).

Questions people ask

What is aberration correction in TEM?

A set of multipole lenses (hexapoles, or quadrupoles and octupoles) placed with the objective lens. They add a negative spherical aberration that cancels the lens's positive Cs. The first working corrected TEM (Haider et al. 1998) used hexapoles; the first sub-ångström STEM probe (Batson, Dellby and Krivanek 2002) used quadrupoles and octupoles.

What is the difference between point resolution and information limit?

Point resolution is the finest detail that transfers with one sign at Scherzer defocus, so the image can be read directly: 0.66(Csλ3)1/4. The information limit is the finest detail that transfers at all, set by the temporal envelope. At 200 kV with Cs 1.2 mm they are 0.245 and 0.139 nm. A corrector closes the gap by moving the point resolution below the information limit.

What is Scherzer defocus?

The underfocus that balances defocus against Cs so that sinχ stays near −1 over the widest band of frequencies: Δf = −1.2√(Csλ). It is −65.8 nm for Cs 1.2 mm at 200 kV. In that band atom columns in a thin crystal look dark.

Why can't a round electron lens be corrected by shaping it better?

Scherzer's theorem (1936): a round, static, space-charge-free lens always has positive spherical and chromatic aberration. To get a negative Cs you must break one of those conditions; practical correctors break round symmetry with multipoles.

What is a Zemlin tableau?

A set of diffractograms (Fourier transforms of images of thin amorphous carbon) recorded with the beam tilted in several directions. How the Thon rings change with tilt gives defocus, astigmatism, coma, threefold astigmatism and Cs. Correctors are tuned from such tableaus in TEM mode, and from Ronchigrams in STEM mode.

What is the difference between a probe corrector and an image corrector?

A probe corrector sits above the specimen and corrects the lens that forms the STEM probe. An image corrector sits below it and corrects the lens that forms the TEM image. The optics are the same; a double-corrected microscope has both.

Why does chromatic aberration matter more after Cs correction?

Once Cs is removed, the temporal envelope from Cc and the energy spread is the next limit. A monochromator (smaller ΔE), a lower Cc, or a Cc corrector moves it. This matters most at low voltage, where ΔE/E is larger.

What is a Ronchigram used for?

It is the shadow image of a thin amorphous film seen in the convergent beam at the detector plane. Its flat central region, the "sweet spot", shows the angle over which the phase is flat, which sets the probe aperture. Probe correctors are tuned by fitting aberrations to Ronchigrams.

Related: SAED pattern simulator (HRTEM tableau and STEM images with the same lens model) · Reading an HRTEM FFT · HRTEM lattice tool · STEM detectors guide · Quantitative HAADF · 4D-STEM basics · EELS guide (energy spread) · Beam damage · TEM contrast guide

References

Show the 11 references
  1. O. Scherzer, Über einige Fehler von Elektronenlinsen, Zeitschrift für Physik 101, 593 to 603 (1936). doi:10.1007/BF01349606. The theorem that round lenses have positive Cs and Cc.
  2. O. Scherzer, Sphärische und chromatische Korrektur von Elektronen-Linsen, Optik 2, 114 to 132 (1947). How to correct them with non-round fields.
  3. O. Scherzer, The theoretical resolution limit of the electron microscope, Journal of Applied Physics 20, 20 to 29 (1949). doi:10.1063/1.1698233. Scherzer defocus.
  4. M. Haider, S. Uhlemann, E. Schwan, H. Rose, B. Kabius and K. Urban, Electron microscopy image enhanced, Nature 392, 768 to 769 (1998). doi:10.1038/33823
  5. P. E. Batson, N. Dellby and O. L. Krivanek, Sub-ångstrom resolution using aberration corrected electron optics, Nature 418, 617 to 620 (2002). doi:10.1038/nature00972
  6. O. L. Krivanek, N. Dellby and A. R. Lupini, Towards sub-Å electron beams, Ultramicroscopy 78, 1 to 11 (1999). The Cn,m notation.
  7. S. Uhlemann and M. Haider, Residual wave aberrations in the first spherical aberration corrected transmission electron microscope, Ultramicroscopy 72, 109 to 119 (1998). The A1, B2, A2 notation and the π/4 tolerances.
  8. F. Zemlin, K. Weiss, P. Schiske, W. Kunath and K.-H. Herrmann, Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms, Ultramicroscopy 3, 49 to 60 (1978).
  9. R. Erni, Aberration-Corrected Imaging in Transmission Electron Microscopy, 2nd ed., Imperial College Press (2015): chapters on the aberration function, correctors, CTF and STEM probes.
  10. E. J. Kirkland, Advanced Computing in Electron Microscopy, 3rd ed., Springer (2020): the CTF with envelopes, the STEM probe and its optimum aperture (α = 1.41(λ/Cs)1/4, Δf = −0.87√(Csλ)).
  11. D. B. Williams and C. B. Carter, Transmission Electron Microscopy, 2nd ed., Springer (2009): chapter 6 (lens aberrations, disc of least confusion) and chapter 28 (the CTF, point resolution and information limit). Si scattering factors: L.-M. Peng, G. Ren, S. L. Dudarev and M. J. Whelan, Acta Crystallographica A 52, 257 to 276 (1996).
Cite this page: Tripathy, Manisha. “Aberration Correction Lab.” untethered atom, 2026, https://untetheredatom.com/tem/aberration-correction.
BibTeX 
@misc{tripathy2026aberrationcorrection,
  author = {Tripathy, Manisha},
  title  = {Aberration Correction Lab},
  year   = {2026},
  howpublished = {\url{https://untetheredatom.com/tem/aberration-correction}},
  note   = {Interactive web tool}
}
Last updated 24 September 2026.