X-Rays and Bragg Scattering

A monochromatic X-ray beam comes in horizontally from the left and hits a two-dimensional crystal. A detector ring around the crystal records the scattered intensity as a function of the scattering angle \(2\theta\), measured from the forward direction (positive = upwards, i.e. anticlockwise). Panel (c) is that ring cut open at \(2\theta = \pm 180^\circ\) and unrolled into a straight line. A peak appears whenever a reciprocal lattice vector \(\mathbf G_{hk}\) lies on the Ewald circle, which is the same as the Bragg condition \(2d\sin\theta = n\lambda\) for one family of lattice planes. Rotate the crystal to bring different reflections into the Bragg condition. Hover over a peak to see which planes it comes from.

2D crystal diffraction: Bragg peaks vs scattering angle Rotate the crystal, then hover over a peak

Crystal structure

Square Rectangular (b = 1.5 a) Square, A–B basis (checkerboard) Hexagonal (triangular) Honeycomb, A–B basis (graphene / hBN) Oblique (b = 1.25 a, γ = 75°)

Crystal rotation φ

Wavelength λ

Crystal size

Scattering strength f_B / f_A

Rotation speed

⟳ Rotate Powder (average over all orientations) Log intensity

(a) Real space: crystal, incident beam from the left, scattered beams, and the detector ring with the measured intensity drawn outwards

(b) Reciprocal space: lattice points G_hk and the Ewald circle

(c) The detector ring from (a), cut at ±180° (back towards the source) and unrolled; the colour band matches the ring. Hover over (or tap) a peak here or on the ring.

Things to try

1. A single crystal rarely diffracts. Pick the square lattice and press Rotate. For most orientations the detector sees nothing. A peak flashes up only when a reciprocal lattice point crosses the Ewald circle, i.e. when some family of planes makes exactly the Bragg angle with the beam. This is the rotating-crystal method.
2. Which planes? Hover over a peak. The reflecting planes light up in the crystal, and the tooltip checks \(2d\sin\theta = n\lambda\). Peaks such as \((2\,0)\) are the second order of the \((1\,0)\) planes.
3. Extinctions. Choose Square, A–B basis and set \(f_B/f_A = 1\). Every reflection with \(h + k\) odd disappears (hollow circles in the Ewald panel), because the lattice is really a smaller, centred square lattice. Lower \(f_B\) and these "superlattice" peaks come back with \(|F|^2 \propto (f_A - f_B)^2\). Turn on Log intensity to see the weak ones.
4. Honeycomb. With \(f_B = f_A\) (graphene), some reflections cancel. With \(f_B \neq f_A\) (hBN) they reappear.
5. Powder diffraction. Tick Powder. Averaging over all orientations, every reflection with \(|\mathbf G| \le 2k\) shows up at once, at \(2\theta = 2\arcsin(\lambda / 2d)\). That is why powder patterns are easy to measure but lose the orientation information. Hover to see which reflections overlap in one ring.
6. Wavelength and crystal size. If \(\lambda > 2d_{\max}\) there are no Bragg peaks at all, because the Ewald circle is too small to reach any reciprocal lattice point. Shrinking the crystal makes the reciprocal lattice spots larger. The peaks get broader and are easier to hit (Scherrer broadening).