Converged illumination turns each spot into a disc with fringes, and the fringe positions give the thickness in nanometres.
Companion to the SAED back-focal-plane schematic. Same optics, one deliberate change (converge the illumination), and each Bragg spot opens into a disc that carries the rocking curve, which is what encodes the thickness. When you have a real pattern in hand, the CBED Thickness Tool runs this whole measurement on your own image, in the browser.
1 · What changes between SAED and CBED
SAED: parallel illumination
Area chosen by the SA aperture (µm-scale, in the image plane).
Pattern = sharp points; thickness broadens relrods and redistributes intensity, but hands you no fringes to read t from.
CBED: a focused cone
Area = the probe itself (nm-scale), no SA aperture needed.
Each incident direction lands at its own point: edge of cone → edge of disc (follow the colors). A disc is a map of incident angle, i.e. of the deviation parameter s.
In SAED a converged beam is a mistake; in CBED it is the whole point.
2 · Two-beam CBED: the disc reads out the rocking curve
Tilt to a two-beam condition (one strong hkl). Across the disc, s varies linearly, so the disc displays Ihkl(s): Kossel–Möllenstedt fringes. Their positions depend on thickness t and extinction distance ξg. Drag the sliders.
Measure the dark-fringe distances xi and the disc separation X straight off the recording: the ratio xi/X makes the method self-calibrating (camera length cancels). The plot below updates live from the fringes above.
Reading the plot
Each dark fringe gives one point. Integers ni are consecutive but n₁ is unknown: try n₁ = 1, 2, 3… and keep the assignment that gives a straight line with negative slope.
Intercept → t (extrapolation to 1/n² = 0). Slope → ξg: check it against the tabulated value, a good consistency test for your n-assignment.
Quick estimate before any fitting: number of dark fringes ≈ t / ξg.
Simulated “measurements”
4 · Bench procedure & pitfalls
At the microscope
Eucentric height, image in focus; then converge the beam to a probe on the exact spot whose thickness you want (spot size up, C2 aperture chosen so α ≈ 0.3–0.7 θB: discs well separated).
Tilt to a clean two-beam condition: one strong hkl, everything else weak. Navigate with Kikuchi bands; the bright band should sit in the hkl disc.
DIFF mode. Diffraction focus sharpens the disc edges (same intermediate-lens physics as the SAED schematic); probe focus on the specimen sharpens the fringes.
Energy-filter (zero-loss) if available: inelastic background is what usually washes fringes out.
Record; measure X and each dark-fringe distance xi; apply the boxed formulas.
Pitfalls
Thickness is measured along the beam at the probe position. If the lamella is tilted for two-beam, correct geometrically if you need foil-normal thickness.
Working range ≈ 1–5 ξg for this graphical method (clean two-beam conditions assumed): thinner → fewer than ~3 fringes (no reliable fit; move to a thicker spot or a larger-ξ reflection); much thicker → absorption smears the fringes.
Avoid bent or strained regions: bending rotates s across the probe and warps the fringes.
Deviating from two-beam (extra strong beams) breaks the simple si² relation.
And the flip side of the SAED rule: parallel illumination cannot give you t this way (no cone, no rocking-curve map, no K–M fringes). Thickness then needs a different route, such as the EELS log-ratio method.
Defaults here: λ = 2.51 pm (200 kV), d = 0.2025 nm (Al 200), so θB ≈ 6.2 mrad. Swap in your own d, λ and tabulated ξg.
Companion artifacts: bfp-diffraction-focus (why the diffraction pattern lives in the BFP and how the focus knob works) and the SAED indexing guide in this project. Same column, same knobs: the only new decision in CBED is the convergence angle.
References
Show the 6 references
P. M. Kelly, A. Jostsons, R. G. Blake and J. G. Napier, The determination of foil thickness by scanning transmission electron microscopy, Phys. Stat. Sol. (a) 31, 771-780 (1975): the (si/ni)2 against (1/ni)2 plot.
S. M. Allen, Foil thickness measurements from convergent-beam diffraction patterns, Phil. Mag. A 43, 325-335 (1981): the extended method and how the unknown first fringe order is handled.
D. B. Williams and C. B. Carter, Transmission Electron Microscopy: A Textbook for Materials Science, 2nd ed., Springer (2009): the chapters on obtaining CBED patterns and on using CBED (Kossel-Möllenstedt fringes and the thickness measurement).
J. C. H. Spence and J. M. Zuo, Electron Microdiffraction, Plenum (1992): the two-beam rocking curve inside a CBED disc and thickness determination.
P. B. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley and M. J. Whelan, Electron Microscopy of Thin Crystals, 2nd ed., Krieger (1977): the two-beam rocking curve Ig(s) that the disc displays.
W. Kossel and G. Möllenstedt, Elektroneninterferenzen im konvergenten Bündel, Ann. Phys. (Leipzig) 36, 113-140 (1939): the first convergent-beam patterns.
Cite this page: Tripathy, Manisha. “CBED vs SAED & Lamella Thickness from K–M Fringes.” untethered atom, 2026, https://untetheredatom.com/tem/cbed-thickness.
BibTeX
@misc{tripathy2026cbedvssaedlamellathickne,
author = {Tripathy, Manisha},
title = {CBED vs SAED & Lamella Thickness from K–M Fringes},
year = {2026},
howpublished = {\url{https://untetheredatom.com/tem/cbed-thickness}},
note = {Interactive teaching resource}
}
