⚗️ Materials Science · Solid State Physics
📅 March 2026⏱ 12 min🟡 Intermediate

Crystal Structure: How Atoms Arrange Themselves

Diamond and graphite are both pure carbon, yet one is the hardest natural material and the other is soft enough to write with. The difference lies entirely in how the carbon atoms arrange themselves — the crystal structure. Understanding crystallography unlocks materials design from semiconductors to jet engine turbine blades.

1. Crystal Lattices and Unit Cells

A perfect crystal is an infinite, periodic arrangement of atoms (or groups of atoms) in 3D space. The lattice is the abstract mathematical framework — an infinite set of points with identical surroundings. The basis is the atom (or group of atoms) placed at each lattice point. Crystal = Lattice + Basis.

Unit cell vectors: A unit cell is the smallest repeating parallelepiped. Defined by lattice vectors: a, b, c (lengths) α, β, γ (angles between them) Atomic packing factor (APF): APF = (volume of atoms in unit cell) / (volume of unit cell) Hard sphere model: atoms touch along close-packed directions. Example, FCC (face-centred cubic): Atoms/cell: 8 × (1/8) + 6 × (1/2) = 4 Atom radius: r = a√2/4 (touch along face diagonal) APF = 4 × (4/3)πr³ / a³ = π/(3√2) ≈ 0.740 ← most efficient packing Dense random packing of spheres: ≈ 0.637

2. Bravais Lattices and Crystal Systems

Auguste Bravais (1848) proved that all possible periodic 3D lattices belong to exactly 14 distinct types, grouped into 7 crystal systems:

7 Crystal Systems with lattice parameter constraints: ┌─────────────┬──────────────────────────────────────────────────────┐ │ Cubic │ a = b = c, α = β = γ = 90° │ │ Tetragonal │ a = b ≠ c, α = β = γ = 90° │ │ Orthorhombic│ a ≠ b ≠ c, α = β = γ = 90° │ │ Rhombohedral│ a = b = c, α = β = γ ≠ 90° │ │ Hexagonal │ a = b ≠ c, α = β = 90°, γ = 120° │ │ Monoclinic │ a ≠ b ≠ c, α = γ = 90° ≠ β │ │ Triclinic │ a ≠ b ≠ c, α ≠ β ≠ γ (most general) │ └─────────────┴──────────────────────────────────────────────────────┘ 14 Bravais lattice types within these systems: Cubic: SC (simple), BCC (body-centred), FCC (face-centred) Tetragonal: Simple, Body-centred etc. Symmetry elements: Point group: rotational symmetry operations (32 crystallographic point groups) Space group: point symmetry + translational symmetry (230 space groups) Every crystal belongs to one of the 230 space groups.

3. Common Crystal Structures in Metals

4. Miller Indices and Crystallographic Planes

Miller index procedure: 1. Identify where the plane intercepts the three crystallographic axes. 2. Take reciprocals of the intercepts. 3. Clear fractions to smallest integers → (h k l) Example: Plane intercepts at a/1, b/2, c/3 → reciprocals: 1, 1/2, 1/3 → multiply by 6: 6, 3, 2 → Miller index (6 3 2) Negative intercept: use overbar notation: (1 0 -1) = (1 0 1̄) Key planes in cubic: (100): cube face (cleavage plane in rock salt) (110): diagonal face (111): octahedral plane (close-packed in FCC — slip plane) Direction indices [u v w]: Direction along vector ua + vb + wc [100] = edge direction, [110] = face diagonal, [111] = body diagonal Families: {hkl} = all equivalent planes by cubic symmetry (e.g., {100} includes (100),(010),(001)) ⟨uvw⟩ = all equivalent directions

5. X-Ray Diffraction and Bragg's Law

William Lawrence Bragg (1913) derived the condition for constructive interference of X-rays diffracting from crystal planes — the primary tool for determining crystal structure:

Bragg's Law: nλ = 2d·sin(θ) λ = X-ray wavelength (typical: 0.05–0.25 nm; Cu Kα = 0.1542 nm) d = interplanar spacing (lattice planes hkl) θ = glancing angle (angle between X-ray beam and crystal plane) n = diffraction order (1, 2, 3 ...) Interplanar spacing for cubic system: d_hkl = a / √(h² + k² + l²) For a = 0.287 nm (α-Fe, BCC), d(110) = 0.287/√2 ≈ 0.203 nm Bragg angle: θ = arcsin(λ/2d) = arcsin(0.1542/0.406) ≈ 22.3° Powder diffraction (Debye-Scherrer method): Crystallite size estimator (Scherrer equation): τ = Kλ / (β·cos θ) K ≈ 0.94, β = peak full width at half maximum Applications: nanocatalyst sizing, thin film characterisation
Structure determination of DNA (1953): Rosalind Franklin's X-ray diffraction photo 51 — a single exposure of B-DNA fibres — showed a characteristic X-pattern (helical diffraction) and systematic absences indicating a double helix with a 3.4 Å rise per base pair and 34 Å pitch. Watson and Crick used this data (and Franklin's unit cell measurements published by Bragg's group) to build the correct double-helix model. Franklin's crystallographic data was the decisive quantitative input.

6. Crystal Defects

Real crystals are never perfect. Defects — departures from the perfect periodic arrangement — profoundly affect mechanical, electrical, and optical properties:

Hall-Petch relation (grain boundary strengthening): σ_y = σ₀ + k_y · d^{-1/2} σ_y = yield strength d = average grain diameter k_y = Hall-Petch slope (material constant) Finer grains → more grain boundaries → harder for dislocations to pass → higher yield strength. Nanocrystalline metals (d < 100 nm): strength 5-10× coarse-grained Amorphous metals: no grain boundaries → highest strength but brittle

7. Structure–Property Relationships