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…
Bragg law (1913)
W H Bragg + W L Bragg 1913 (Nobel 1915): n lambda = 2 d sin(theta); reciprocal-lattice formulation Laue equations Delta-k = G; basis of…
Space groups (Fedorov-Schoenflies)
Fedorov 1891 + Schoenflies 1891: 230 crystallographic space groups in 3D; classification by point-group + Bravais-lattice + screw/glide…
Phase problem (direct methods)
Hauptman-Karle 1953-1985 (Nobel 1985) direct methods: Sayre's equation + tangent formula + Karle-Hauptman determinants exploit positivity +…
Rietveld refinement (1969)
Rietveld 1969 powder-pattern refinement: full-profile least-squares minimizes sum w_i (y_i,obs - y_i,calc)^2; quantitative phase analysis;…
Electron density Fourier synthesis
rho(r) = (1/V) sum_h F(h) exp(-2 pi i h.r); Patterson 1934 |F|^2 map; difference-Fourier maps localize ligands; basis of macromolecular…
Quasi-crystals (Shechtman 1984)
Shechtman 1984 (Nobel 2011) Al-Mn icosahedral diffraction with 5-fold symmetry; Penrose-tile aperiodic order; long-range order without…
Patterson function (1934)
A L Patterson 1934 vector-map P(u)=∫|F|² e^{-2πi u·k}; modern heavy-atom-method + macromolecular phasing foundation.
Direct methods (Sayre 1952)
D Sayre 1952 + Hauptman-Karle 1953 (Nobel 1985) direct-methods phase-determination; modern SHELXD + ShelXS small-mol crystals.
MAD phasing (Hendrickson 1991)
W Hendrickson 1991 multi-wavelength-anomalous-diffraction; modern Se-Met SAD + 100k+ macromolecular structures since.
Electron crystallography (Henderson 1990)
R Henderson 1990 (Nobel 2017) bacteriorhodopsin 3.5Å electron-crystallography; modern cryo-EM revolution Frank+Dubochet.
MicroED (Gonen 2013)
T Gonen 2013 micro-electron-diffraction; modern small-molecule MicroED Gruene-Hovestreydt 2018 sub-mg structures.
Charge flipping (Oszlanyi 2004)
G Oszlanyi-A Süto 2004 charge-flipping algorithm; modern superflip + ab-initio phase recovery from intensities alone.