X-ray Bragg diffraction · crystallography

You cannot see atoms — so how do you measure the distance between the ions in a grain of salt, and could that atomic ruler possibly weigh out a whole mole?

▶ Run the simulationSee the measured result

Units: angstrom (NaCl (200) interplanar spacing)

How the lab tests it

Forward-model the founding crystallography experiment (Bragg, 1913). Shine monochromatic X-rays (Cu Kα, λ=1.5406 Å) on a stack of N=48 parallel atomic planes spaced d apart; the path difference between waves reflected off adjacent planes is 2d·sinθ, so the diffracted intensity I(θ)=sin²(Nφ/2)/sin²(φ/2) (φ=2π·2d·sinθ/λ) spikes only at the Bragg angles 2d·sinθ=mλ. Generate the diffractometer trace (d enters ONLY here, as the physical crystal), find the principal-maximum angles by threshold + parabolic sub-grid refinement, and invert each via d=mλ/(2·sinθ_m) — a formula that never contains d. Then set the cubic cell edge a=2d and recover N_A=Z·M/(ρ·a³) from the macroscopic density. Sweep the crystal spacing as a perturbation and run the transmission-grating law d·sinθ=mλ as a falsification control. ?world=bragg.

What it checks

Bragg's law 2d·sinθ=mλ — the NaCl (200) interplanar spacing d=2.820 Å recovered to ~1e-4 purely from the three peak angles (15.85°, 33.11°, 55.03°; the 4th order forbidden, sinθ>1), and tracking the input across d∈[2.0,4.0] Å. The result is decisive because the factor of 2 is physical: inverting the SAME peaks with the transmission-grating law d·sinθ=mλ (no ×2) recovers 2d=5.64 Å — wrong by exactly 2× — so only the pair-of-planes path difference is self-consistent. And the atomic ruler weighs a mole: the recovered cell edge a=2d=5.6402 Å feeds N_A=Z·M/(ρ·a³)=6.02e23 to within 0.07% of CODATA — the macroscopic-to-atomic bridge. Distinct from ?world=debroglie's electron diffraction (surface d·sinθ=λ, recovering the electron wavelength): Bragg reflects X-rays off 3-D volume planes to recover the lattice itself. The lab's first crystallography world.

Bragg's law calculator

Solve Bragg's law for whichever quantity you're missing. Change any three boxes and the fourth is recomputed — the same relation the simulation above recovers from raw diffraction peaks.