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 \times (1/8) + 6 \times (1/2) = 4 Atom radius: r = a\sqrt2/4 (touch along face diagonal) APF = 4 \times (4/3)πr³ / a³ = π/(3\sqrt2) \approx 0.740 \leftarrow most efficient packing Dense random packing of spheres: \approx 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 \neq c, α = β = γ = 90° │ │ Orthorhombic│ a \neq b \neq c, α = β = γ = 90° │ │ Rhombohedral│ a = b = c, α = β = γ \neq 90° │ │ Hexagonal │ a = b \neq c, α = β = 90°, γ = 120° │ │ Monoclinic │ a \neq b \neq c, α = γ = 90° \neq β │ │ Triclinic │ a \neq b \neq c, α \neq β \neq 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

FCC (Face-Centred Cubic): Al, Cu, Ni, Au, Ag, γ-Fe. APF = 0.74. 12 nearest neighbours. Slip system: {111}⟨110⟩ — 12 equivalent slip systems → high ductility. Used in aerospace alloys.
BCC (Body-Centred Cubic): α-Fe, W, Mo, Cr, V. APF = 0.68. 8 nearest neighbours. 48 slip systems but narrower slip planes → higher strength, lower ductility than FCC. BCC metals often show ductile-to-brittle transition at low temperatures.
HCP (Hexagonal Close-Packed): Ti, Zn, Mg, Co. APF = 0.74 (same as FCC). 12 nearest neighbours. Fewer slip systems than FCC → less ductile. Ti alloys in jet engines, Mg alloys in automotive lightweight structures.
Diamond cubic: C (diamond), Si, Ge. APF = 0.34 (very low — strong directional covalent bonds). Tetrahedral bonding. Si is the foundation of all semiconductor devices.
NaCl structure (rock salt): Two interpenetrating FCC lattices. Na⁺ and Cl⁻ ions alternate. Common ionic crystals: NaCl, MgO, FeO.

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:

Point defects (0D): Vacancies (missing atom), interstitials (extra atom in gap), substitutional impurities. Vacancies are thermodynamically inevitable at T > 0 K (equilibrium vacancy concentration n/N = exp(−E_f/k_BT)). Carbon interstitials in BCC iron = steel.
Line defects — dislocations (1D): Edge and screw dislocations. Characterised by Burgers vector b. Plastic deformation occurs by dislocation motion along slip planes. See the companion article on dislocations.
Planar defects (2D): Grain boundaries (interfaces between differently oriented crystals), stacking faults (wrong stacking sequence, e.g., ABABAB... in HCP vs ABCABC in FCC), twin boundaries.
Volume defects (3D): Voids (clusters of vacancies), precipitates (second-phase particles), inclusions.

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

Mechanical: FCC metals are ductile (many slip systems). HCP metals anisotropic. Covalent crystals (diamond, SiC) extremely hard. Ionic crystals brittle (electrostatic repulsion when planes shear). Precipitation hardening (Al 2xxx/7xxx series): coherent nanoscale precipitates block dislocations → 3× strength increase.
Electrical: Full valence band → insulator (diamond). Partially filled or overlapping → metal. Small band gap (Si 1.1 eV, Ge 0.67 eV) → semiconductor. Dopant atoms substitute lattice sites → donor/acceptor levels in gap.
Optical: Anisotropic crystals (calcite, quartz) are birefringent — refractive index depends on polarisation direction. Used in optical waveplates and polarisers.
Piezoelectric: Non-centrosymmetric crystal classes (20 of 32 point groups) develop electric polarisation under mechanical stress. Quartz (SiO₂), BaTiO₃, PZT — oscillators, sensors, actuators.