Fourier duality
/ FOOR-ee-ay /
Fourier duality is the deep reason the reciprocal lattice exists and behaves the way it does. A Fourier transform is a mathematical machine that takes a pattern and reports what regular repeats hide inside it — the way a prism splits white light into its colours, or a music app shows the bass and treble bars in a song. Do that operation on a crystal, and out comes its reciprocal lattice. In one sentence: the reciprocal lattice IS the Fourier transform of the direct lattice.
The transform obeys a small set of rules that between them explain every property of reciprocal space. Repeating patterns transform into sharp spikes (which is why a periodic crystal gives sharp diffraction spots, not a smear). The transform of a lattice is itself a lattice — the reciprocal one. And, crucially, the transform stretches what is small and shrinks what is large: a pattern that repeats over a short distance shows up at a high frequency, far out in reciprocal space. This last rule is the inverse-size relationship, and it is not a coincidence but a theorem about Fourier transforms. Diffraction is nature performing this transform for you: the scattered wave amplitude is, to a good approximation, the Fourier transform of the crystal's electron density.
This duality is what makes crystallography possible AND what makes it hard. Possible, because measuring the diffraction pattern is measuring the Fourier transform, so an inverse transform should hand you the structure back. Hard, because a detector records only the INTENSITY of each spot (the amplitude squared) and throws away the PHASE of the Fourier components — and you cannot invert a Fourier transform without the phases. That missing information is the phase problem, the central obstacle in structure determination, and it exists precisely because diffraction gives you the Fourier magnitudes but not the Fourier phases.
Take a row of atoms spaced 3 angstrom apart. Its Fourier transform is a row of reciprocal points spaced 1/3 = 0.333 angstrom^-1 apart. Halve the real spacing to 1.5 angstrom and the reciprocal spacing doubles to 0.667 angstrom^-1 — the transform reads out the repeat rate directly.
A tighter real repeat becomes a wider reciprocal spacing: the signature move of a Fourier transform.
Diffraction gives the Fourier MAGNITUDES but not the PHASES, so you cannot simply invert it to see the structure — that lost phase is the phase problem, not a limitation of your instrument.