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?
▶ Launch the interactive simulationForward-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.
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.