Quantitative science of crystal structure: space groups, Bravais lattices, Bragg diffraction. Since 1912 (von Laue, Bragg) the definitive method for atomic-resolution structure determination — the substrate of structural biology, materials…
crystallography
Bravais lattice
Infinite set of points R = n₁a₁+n₂a₂+n₃a₃ (n_i ∈ ℤ) whose neighbourhood appears identical. Exactly 14 distinct Bravais lattices in 3D,…
Unit cell
Parallelepiped defined by a₁, a₂, a₃ that tiles space by translation. Six parameters (a,b,c,α,β,γ) with seven-crystal-system constraints;…
Space group
Symmetry group combining lattice translations with point-group operations (rotations, reflections, inversions) plus screw axes and glide…
Bragg's law
Coherent constructive interference of X-rays off parallel lattice planes of spacing d at incidence angle θ: nλ = 2d sin θ. Gives peak…
Reciprocal lattice
Lattice of wavevectors b_i with a_i·b_j = 2πδ_ij. Crystallographic diffraction peaks occur when scattering vector Q = G for some…
X-ray diffraction (XRD)
Elastic scattering of X-rays by the electron density ρ(r); observed intensity I(Q) ∝ |F(Q)|² with structure factor F(Q) = Σ_j f_j…
Miller indices (hkl)
Triple (h,k,l) of integers identifying a lattice plane with intercepts (a/h, b/k, c/l). Associated plane spacing 1/d² = h²/a² + k²/b² +…
Structure factor and phase problem
Measured diffraction gives |F(Q)| only; phase φ(Q) is lost. Reconstruction of ρ(r) requires phase determination: direct methods…
Rietveld refinement
Full-profile fitting of powder diffraction patterns: least-squares minimisation of Σ_i w_i (y_i^obs – y_i^calc)². Extracts lattice…
Electron crystallography / microED
Diffraction and imaging of crystals with electron beams (λ~2 pm at 300 kV). Much stronger scattering than X-rays enables sub-micron…
Bravais classification: exactly 14 lattice classes in 3D (crystallography)
Chemistry application of L1 bloch-theorem and brillouin-zone. Bravais' 1850 classification theorem enumerates the translational-symmetry…
Systematic extinction rules from structure factor F(hkl) (crystallography)
Chemistry application of L1 bragg-law and phonon-physics. Systematic extinctions in X-ray, electron, and neutron diffraction arise from…