CBED: from discs to lamella thickness

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.

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 only smears intensity — you can't read t from it.

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 I_hkl(s): Kossel–Möllenstedt fringes. Their positions depend on thickness t and extinction distance ξ_g. Drag the sliders.

Convergence semi-angle α = mrad (C2 aperture)

Lamella thickness t = nm

Extinction distance ξ_g = nm (from tables)

3 · Kelly–Allen extrapolation: fringe positions → thickness

s_i = ^x_i⁄_X · ^λ⁄_d² s_i² = ^n_i²⁄_t² − ¹⁄_{ξ_g²} → plot ( s_i/n_i )² vs ( 1/n_i )² : intercept = 1/t², slope = −1/ξ_g²

Measure the dark-fringe distances x_i and the disc separation X straight off the recording — the ratio x_i/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 n_i 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 → ξ_g: check it against the tabulated value — a good consistency test for your n-assignment.
Quick estimate before any fitting: number of dark fringes ≈ t / ξ_g.

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 θ_B: 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 x_i; 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 ξ_g: 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 s_i² relation.
And the flip side of the SAED rule: parallel illumination can never give you t — no cone, no rocking-curve map, no fringes.

Defaults here: λ = 2.51 pm (200 kV), d = 0.2025 nm (Al 200), so θ_B ≈ 6.2 mrad. Swap in your own d, λ and tabulated ξ_g.

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.