Lab 16 — Crystallography

Crystals & Diffraction — The Reciprocal World

Shine X-rays or electrons on a crystal and a pattern of Bragg spots appears on the screen. That pattern is the 2D Fourier transform of the crystal’s electron density — and what a detector measures is its intensity |F(q)|². Left is the real-space crystal; right is its diffraction. Because the Fourier transform commutes with rotation, spinning the crystal spins the pattern in lock-step. Widen the spacing d and the spots move closer — that is “reciprocal space.”

F(\mathbf{q})=\sum_j f_j\,e^{\,i\,\mathbf{q}\cdot\mathbf{r}_j}\qquad I(\mathbf{q})=\bigl|F(\mathbf{q})\bigr|^2\qquad 2d\sin\theta = n\lambda

Change the lattice (square/hex) and basis and the spots rearrange. Switch to “2 atoms” and structure-factor extinctions thin them out. “Discard phase” lets you feel the phase problem.

Lattice spacing dd = 5.7 Å

Lattice

Basis (motif)

Phase

Real-space spacing—

X-ray crystallography is this inverse problem — measure the diffracted intensities |F|² and Fourier-transform back to the electron density (the atomic positions). But only the intensities are measured; the phase is lost (the phase problem). Inverse-transforming intensities alone yields not an image of the atoms but a map of interatomic vectors — the autocorrelation, or Patterson function. In practice the phases are recovered by heavy-atom, molecular-replacement, or direct methods. The same reciprocal-space idea underlies electron diffraction and MRI (k-space). Even DNA’s double helix was read from the spot pattern of Franklin’s diffraction image, “Photo 51.”