How to read zone-axis tilts and Kikuchi lines

Both 3-D lattices share one rotation. Drag either and the other follows, because the reciprocal lattice is bolted to the crystal. The electron beam (green arrow, always vertical: it goes straight down the microscope column and never moves) and the Ewald sphere stay put: you bring the crystal to the beam with the specimen holder, not the other way round. Points lying within a small excitation error of the sphere surface (their relrods touch it) light up and land on the detector. On a zone axis a whole plane of reciprocal-lattice points satisfies this at once. Snap to [001] (square net), [011] (centred-rectangular net) or [111] (hexagonal net) and watch the pattern symmetry announce the orientation. Slide the radius down toward λ ~ d (X-ray-like) and the sphere curls up: only a thin ring of points intersects it, which is why a single still X-ray exposure sees far less of reciprocal space than a 200-kV SAED pattern, where 1/λ ≈ 400 nm⁻¹ makes the sphere effectively a plane.

Real vs reciprocal in one sentence: real space maps where the atoms are (positions, lengths a, d_hkl); reciprocal space maps how the planes repeat (one point g_hkl per plane family, ⊥ to the planes, |g| = 1/d). The two are Fourier transforms of each other, so they scale inversely (drag the lattice-parameter slider: as a grows, the g-net shrinks by exactly 1/a) but they rotate identically, which is why a diffraction pattern measures orientation at all. Big in real space = fine in reciprocal space (and vice versa: small precipitates → broad spots, thin plates → long relrod streaks).

Selecting a plane: type its Miller indices (hkl). The teal sheet is one representative of the {h₁k₁l₁} family, the amber sheet one of {h₂k₂l₂}; the whole family repeats parallel to it with spacing d_hkl = a₀/√(h²+k²+l²): higher indices → finer spacing → longer g. The arrow ĝ ⊥ the planes is the reciprocal-space direction of that family; its angles to the crystal axes [100], [010], [001] follow cos θᵢ = hᵢ/√(h²+k²+l²), and the angle φ between two plane families is the angle between their normals: for cubic, cos φ = (h₁h₂+k₁k₂+l₁l₂)/(√N₁√N₂).

Zone axis and beam: the two families intersect along a common direction: the zone axis B = g₁ × g₂ (Weiss zone law hu+kv+lw = 0 holds for both). Both families diffract simultaneously only when the beam ∥ B: press Align and watch the crystal tilt until B meets the fixed beam, exactly what you do with the double-tilt holder while walking along Kikuchi bands. On zone, both spot rows appear and the angle between the rows equals φ, the "ratio AND angle" check used to index SAED patterns. Drag the crystal away and the rows die as their relrods leave the Ewald plane.

Grain A (teal) and grain B (copper) share the same lattice, rotated by Δθ. Where the nets nearly coincide the spots split: a low-angle boundary (≲10–15°, physically a dislocation array) shows paired/arced spots. Beyond ~15° you resolve two independent nets: a general high-angle boundary. The twin toggle mirrors the net instead: reflection across a mirror plane, the 2-D analogue of the FCC Σ3, 60°⟨111⟩ twin whose ⅓{422} extra spots are one of the "five suspects" in the SAED indexing guide.

Each grain's reciprocal-lattice point g_hkl sits at radius 1/d_hkl; the ensemble of grain rotations sweeps each point around that circle. Random orientations populate it uniformly → Debye–Scherrer rings. Slide up the texture control and the same grains cluster about a preferred orientation φ₀: rings collapse into arcs whose azimuthal spread is a direct picture of the texture (what a pole figure quantifies). One grain = spot net; a handful = overlapping nets; hundreds = continuous rings.