Interactive Guide · TEM Series

Why only some diffraction spots appear

The sphere is imaginary. The missing spots are very real.

Every diffraction pattern in a transmission electron microscope lives in a strange, inside-out version of the crystal called reciprocal space. Once you can picture it, and the giant sphere that slices through it, diffraction patterns stop being mysterious dot art and start being maps.

1 · A crystal and its mirror world

A crystal is a repeating pattern of atoms. Its diffraction pattern is also a pattern of spots, but the two are related inversely: planes that are far apart in the crystal make spots that are close together in the pattern, and vice versa. Rotate the crystal, and the pattern rotates with it.

Real space: the atoms

Reciprocal space: the diffraction spots

Plane spacing d:

Crystal rotation:

Try it: drag the spacing slider up. The atoms spread out, and the spots crowd in. That inverse dance is the whole idea of reciprocal space: each spot sits a distance 1/d from the center.

2 · The Ewald sphere: which spots light up?

Reciprocal space contains a spot for every set of atomic planes. But your diffraction pattern only shows a few of them. Why those?

The rule is geometric. The electron beam is drawn as an arrow of length 1/λ (one over the electron wavelength), ending at the center spot. Sweep that arrow around and it traces a huge sphere: the Ewald sphere. A reciprocal-space spot appears in your pattern only if the sphere's surface passes through it (or close enough, more on that below).

Electron wavelengths are tiny: about 2.5 picometers at 200 kV (a Talos 200i) and 2.0 pm at 300 kV (a Spectra 300), so the sphere is enormous and its surface is nearly flat where it cuts through the spots. That's why a TEM pattern shows a whole plane of spots at once.

Top: side view of the Ewald sphere slicing reciprocal space. Bottom: the diffraction pattern you'd see on the screen.

Voltage:

Crystal tilt:

Sample thickness:

λ = sphere radius =

Try it: tilt the crystal a fraction of a degree. Watch spots on one side brighten while the other side fades, exactly what happens at the microscope when you rock the specimen. Then raise the voltage: the sphere flattens and more spots satisfy the condition at once.

3 · Relrods: why thin samples are generous

Strictly, the sphere should hit each spot exactly, which almost never happens. Yet real patterns are full of spots. The escape clause: your sample is a thin foil. Squeezing a crystal thin in one direction stretches its reciprocal-space spots into little rods along the beam direction, called relrods. The sphere only has to pass through the rod, not the point.

The miss distance, how far the sphere passes from the spot's center, is the excitation error, s. Small miss, bright spot; big miss, dim spot; beyond the rod, nothing.

Try it: in the simulator above, make the sample thinner. The rods grow, more spots light up, and the pattern gets busier. Thicken it and the condition tightens: fewer, sharper spots. This is why thin areas of your foil give the friendliest patterns.

The rocking curve: what “close enough” is worth

Calling s a miss distance is only half the story. The useful question is how much intensity a given miss costs you, and the answer is a curve you can put numbers on. For a foil of thickness t the kinematical result is

I_g = sin²(πts) / (ξ_g²s²)

which is the sinc-squared shape mentioned above, scaled by the extinction distance ξ_g: a length that says how strongly this particular reflection scatters. Two things follow immediately. The curve has zeros at s = n/t, so a thicker foil has a narrower central peak, which is the relrod statement in reverse. And at s = 0 the formula gives (πt/ξ_g)², which passes 1 as soon as the foil is thicker than about ξ_g/3. An intensity above 1 means more electrons leaving than arrived, so the kinematical model has to be wrong exactly where the spot is brightest.

The two-beam dynamical result repairs it with one substitution: replace s by the effective excitation error s_eff = √(s² + ξ_g⁻²). Nothing else changes. The peak stops diverging, and at exact Bragg the intensity oscillates as sin²(πt/ξ_g), which is where thickness fringes come from.

Driven by the voltage, tilt and thickness sliders above, plus the two below.

Reflection d-spacing:

Extinction distance ξ_g:

Readoutss = w = sξ_g = t/ξ_g = I kinematical = I dynamical =

Try it: set the tilt to zero and read s. It is not zero, because at an exact zone axis every spot except 000 already sits off the sphere by −λg²/2, which is the curvature term. Tilting past that value is what puts one reflection into the two-beam condition, and it is why a two-beam image is never taken down a zone axis. Then drive the thickness up and watch the kinematical curve climb through 1 while the dynamical one stays put.

Scope: one reflection, no absorption, no other beams. ξ_g is a slider here because it depends on the structure factor of the particular reflection and on the voltage: for FCC nickel at 200 kV the 111 extinction distance is about 66 nm and the 220 about 104 nm (Oxford Materials practical 2P11, 2024). The SAED pattern simulator computes it from the structure factor for real materials, and the bend contours page draws this same curve as a surface over a whole tilt plane, which is what a bent foil samples all at once. For the weak-beam case, where you deliberately work at large w, see the weak-beam dark field guide.

Key takeaways

Reciprocal space is the crystal's inverse twin: big spacings → close spots, and rotations carry over directly.
The Ewald sphere (radius 1/λ) decides which spots appear: in the kinematic ideal, only those its surface touches, until relrods (below) relax the rule.
Electron wavelengths are so short that the sphere is nearly flat: you see a whole 2D slice of reciprocal space at once.
Thin foils stretch spots into relrods, relaxing the condition; the excitation error s measures the miss and sets the spot's brightness.
Tilting the crystal steers reciprocal space through the sphere: this is what you're really doing with the goniometer.
The rocking curve puts a number on the miss: intensity falls as sin²(πts)/(ξ_g²s²), with zeros every 1/t, and the kinematical form fails once the foil passes about ξ_g/3.

For the physics-curious: the equations behind the pictures

Bragg's law, λ = 2d sinθ, sets the diffraction angle. In reciprocal space each set of planes becomes a vector g with |g| = 1/d. The incident beam is a wavevector k₀ with |k₀| = 1/λ; diffraction to k is allowed when k − k₀ = g (the Laue condition), which is precisely "g lies on the Ewald sphere."

At 200 kV the relativistic wavelength is λ ≈ 2.51 pm, so the sphere radius is 1/λ ≈ 398 nm⁻¹ while a typical |g| is only ~4–10 nm⁻¹: the sphere is ~50× larger than the pattern you record, hence nearly flat. At 300 kV, λ ≈ 1.97 pm and it's flatter still.

A foil of thickness t gives relrods of length ~2/t. In the kinematic finite-thickness model the diffracted intensity varies as sin²(πt s)/(πs)², the familiar sinc-squared shape, so intensity falls off smoothly with excitation error s and oscillates with thickness; dynamical diffraction reshapes the numbers in thicker foils, but not the geometry.

Cite this page: Tripathy, Manisha. “Reciprocal Space & the Ewald Sphere.” untethered atom, 2026, https://untetheredatom.com/tem/ewald-sphere-guide.

BibTeX

@misc{tripathy2026reciprocalspacetheewalds,
  author = {Tripathy, Manisha},
  title  = {Reciprocal Space & the Ewald Sphere},
  year   = {2026},
  howpublished = {\url{https://untetheredatom.com/tem/ewald-sphere-guide}},
  note   = {Interactive teaching resource}
}

Last updated 23 September 2026.