This sim puts a two-dimensional crystal beside its diffraction pattern. Change the cell on the left and the spots on the right respond, which is how crystallographers get from a pattern to a cell they can never photograph.

What's on screen

• Atoms (amber), one per cell, with one cell outlined. Red is side a, green is side b, and the yellow arc is the angle γ.
• Spots (cyan) on the right, one for every integer pair (h, k). The white cross is (000), the straight-through beam.
• Selected spot (yellow): the line from (000) to the spot. Its set of pale lines appears on the crystal too. Click any spot to select it.
• Readouts: the selected spot's (h, k), its distance |G| from (000) in nm⁻¹, and its row spacing d in Å.

Press ⌖ on a row to see where that control sits in the sim.

Fixed: both panels keep a constant scale, so any motion is a real change. One atom per cell, and every spot is drawn equally bright.

The spots are not a magnified picture of the atoms. The sim builds them from the real cell with one rule: each reciprocal vector b_j has dot product 2π with its own real vector and 0 with the other.

In the plane, for sides a, b and angle γ, this gives:

The spot labelled (h, k) sits at the reciprocal-lattice vector below. Its length is measured in nm⁻¹ because the sim keeps the 2π: reciprocal lengths are 2π divided by a crystal length.

A selected spot has a set of rows: parallel lines of atoms, a distance d apart. They run along t = k a₁ − h a₂.

So G is perpendicular to its rows for every a, b and γ, not just for a square cell. How far out the spot sits is fixed by the spacing of those rows:

The Invariant readout |G| × d reads 6.283 = 2π at every spot. (|G| is in nm⁻¹ and d in Å, so the sim divides the product by 10.)

• a appears only in terms that carry h, and b only in terms that carry k. A spot with h = 0 does not depend on a.
• The shear term −2hk cos γ/(ab) is zero at γ = 90° and whenever hk = 0.
• Away from 90°, a spot on an axis moves outward, since |G| = 2π|h|/(a sin γ) when k = 0. For hk ≠ 0 the sign of hk decides it. At a = b = 4 Å and γ = 75°, (1,1) falls from 22.214 to 19.799 nm⁻¹ while (1,−1) rises to 25.803.

A beam of wave vector k scatters elastically from every atom, so |k′| = |k| = 2π/λ. Waves from all the cells add in phase only when the change in wave vector is a reciprocal-lattice vector:

• Bragg's law is the same condition. |Δk| = 2|k| sin θ and |G| = 2π/d give 2d sin θ = λ, with θ measured from the rows.
• A spot such as (2,0) is the second order of the (1,0) rows. The sim draws it with rows at half the spacing.
• What the sim leaves out: it draws every G equally bright, with no wavelength, beam direction or intensity. A real exposure shows only the G that its λ and orientation reach.

Measure |b₁|, |b₂| and the angle between b₁ and b₂ on a pattern, and the cell follows. The angle is the supplement of γ:

The lengths then give a = 2π/(|b₁| sin γ) and b = 2π/(|b₂| sin γ). Each row of spots leans |90° − γ| off the cell side it belongs to, and matches it only at γ = 90°.

On this plate the angle between b₁ and b₂ is 75°, so γ = 105°. With |b₁| = 10.841 nm⁻¹ the lengths give a = 6 Å and b = 3 Å. A lean the other way would mean γ = 75°.

• Choosing the radiation. Bragg's law needs λ no larger than 2d. X-rays (Cu Kα: 1.54 Å) suit spacings of a few Å. Visible light at 5000 Å cannot resolve a unit cell.
• Surfaces. In low-energy electron diffraction (LEED) the spot pattern is the reciprocal lattice of the surface layer, so its spacings give the surface cell directly.
• DNA, 1952. Franklin and Gosling's Photo 51 was read with the same inversion: its layer-line spacing gave the 34 Å pitch of the helix, and a spot at 3.4 Å gave the rise per base. (A fibre, not a crystal.)

Key points

• The reciprocal basis obeys a_i·b_j = 2πδ_ij, and spot (h, k) sits at G = h b₁ + k b₂.
• G ⟂ rows and |G| d = 2π for every a, b and γ.
• a enters |G| only through terms with h, and b only through terms with k.
• Spots appear where Δk = G, which is Bragg's law 2d sin θ = λ.

Common mix-ups

• "The pattern is a magnified photograph of the crystal." It is the reciprocal lattice. Stretching a moves the (1,0) spot toward the centre.
• "Rows of spots always stay parallel to the cell sides." Only at γ = 90°. Otherwise they lean |90° − γ|, and the angle between b₁ and b₂ is 180° − γ.
• "Changing a moves every spot." Spots with h = 0 do not depend on a.

Things to try on your own

• Set γ = 120° and compare the angle between the lines to (1,0) and (0,1) with 180° − γ.
• Set γ = 75° and compare |G| for (1,1) and (1,−1) with their values at 90°.
• At γ = 90°, select any spot, double a, and check that |G| × d stays at 6.283.