Structure of Materials Notes

Last reviewed 16 Sept 2026 · 10 min read

Levels of structure

Level	Scale	Examples
Atomic/electronic	< 1 nm	Bonding type (metallic, ionic, covalent)
Crystal structure	~ 0.1–1 nm	Arrangement of atoms in lattices (BCC, FCC)
Microstructure	µm to mm (seen by microscopes)	Grains, phases, pores, inclusions
Macrostructure	Visible to naked eye	Weld zones, cracks, aggregate in concrete, wood grain

Properties of materials depend on structure at all these levels.

Bonding and properties

Bond	Materials	Typical properties
Metallic	Metals	Good electrical and thermal conductivity, ductility, lustre
Ionic	Ceramics (MgO, Al₂O₃)	Hard, brittle, high melting point, insulating
Covalent	Diamond, SiC, SiO₂, polymers (chain backbone)	Very hard (network solids), directional bonds, insulating
Secondary (van der Waals, hydrogen)	Between polymer chains, in clays	Weak — low melting points of thermoplastics, plasticity of clays

Crystalline and amorphous solids

Crystalline	Amorphous (non-crystalline)
Long-range periodic arrangement of atoms	Only short-range order
Sharp melting point	Soften over a range (glass transition)
Anisotropic properties possible	Isotropic
Metals, most ceramics, ice, quartz	Glass, many polymers, amorphous silica (e.g. in silica fume)

Crystal lattices

Space lattice — infinite 3D array of points with identical surroundings.
Unit cell — smallest repeating unit, described by edge lengths a, b, c and angles α, β, γ.
Seven crystal systems: cubic, tetragonal, orthorhombic, rhombohedral (trigonal), hexagonal, monoclinic, triclinic.
14 Bravais lattices (combinations with simple, body-centred, face-centred, base-centred arrangements).

Cubic and hexagonal structures

FormulaStructure parameters

Structure	Atoms per unit cell	Coordination number	Relation of a and atomic radius r	Atomic packing factor (APF)	Examples
Simple cubic (SC)	1	6	$a = 2 r$	0.52	Polonium
Body-centred cubic (BCC)	2	8	$a = \frac{4 r}{3}$	0.68	α-iron (ferrite), Cr, W, Mo, V
Face-centred cubic (FCC)	4	12	$a = 22 r$	0.74	Al, Cu, Ni, Ag, Au, Pb, γ-iron (austenite)
Hexagonal close-packed (HCP)	6 (in hexagonal prism cell)	12	$c / a \approx 1.633$ (ideal)	0.74	Zn, Mg, Ti, Co, Cd

APF $= \frac{volume of atoms in unit cell}{volume of unit cell}$

Theoretical density $ρ = \frac{n A}{V _{c} N _{A}}$ ( $n$ = atoms per cell, $A$ = atomic mass, $V_{c}$ = cell volume, $N_{A}$ = Avogadro's number)

FCC metals have many slip systems → generally ductile even at low temperatures (Al, Cu).
BCC metals (e.g. ferritic steels) can show a ductile-to-brittle transition at low temperatures.
HCP metals have fewer slip systems → often less ductile at room temperature.

Miller indices

Notation for crystal planes (hkl) and directions [uvw].

Finding Miller indices of a plane:

Find intercepts on x, y, z axes in units of lattice parameters.
Take reciprocals.
Clear fractions to smallest integers; enclose in parentheses (hkl). Negative indices shown with a bar.

A plane parallel to an axis has intercept ∞ → index 0.
Family of planes {hkl}; family of directions ⟨uvw⟩.
In cubic crystals, direction [hkl] is perpendicular to plane (hkl).
Interplanar spacing (cubic): $d_{hk l} = \frac{a}{h ^{2} + k ^{2} + l ^{2}}$ — used with Bragg's law in X-ray diffraction.
Close-packed planes: {111} in FCC; {110} in BCC (most densely packed).

Polymorphism and allotropy

Polymorphism — same composition, different crystal structures (allotropy for elements).
Iron: α-iron (BCC) stable up to 912 °C; γ-iron (FCC, austenite) 912–1394 °C; δ-iron (BCC) 1394–1538 °C (melting). This transformation makes heat treatment of steels possible.
Carbon: diamond, graphite, fullerenes, graphene.
Silica (SiO₂): quartz, tridymite, cristobalite.

Crystal imperfections (defects)

Type	Dimension	Examples	Effects
Point defects	0-D	Vacancy (missing atom), interstitial (extra atom in voids), substitutional impurity; in ionic crystals Schottky (cation–anion vacancy pair) and Frenkel (ion displaced to interstitial site)	Diffusion, conductivity, solid solution strengthening
Line defects (dislocations)	1-D	Edge dislocation (extra half-plane; Burgers vector ⊥ dislocation line), screw dislocation (Burgers vector ∥ line), mixed	Plastic deformation by slip occurs through dislocation motion; work (strain) hardening by dislocation interaction
Surface (planar) defects	2-D	Grain boundaries, twin boundaries, stacking faults, free surfaces, phase boundaries	Obstruct dislocation movement — strengthening; corrosion sites
Volume defects	3-D	Voids, pores, cracks, inclusions, precipitates	Reduce strength and toughness; stress concentration

Strengthening mechanisms

Grain size refinement — more grain boundaries → higher strength and toughness. Hall–Petch relation: $σ_{y} = σ_{0} + k_{y} d^{- 1/2}$ (smaller grain size d → higher yield strength).
Solid solution strengthening (alloying atoms distort lattice).
Strain (work) hardening (cold working).
Precipitation (age) hardening (fine particles).
Dispersion strengthening, martensitic transformation (quenching steels).

Solid solutions and phase diagrams

Solid solution — atoms of solute dissolved in the crystal of solvent: substitutional (similar atomic sizes — Cu–Ni) or interstitial (small atoms like C, N, H in iron).
Hume–Rothery rules for extensive substitutional solubility: similar atomic radius (within about 15%), same crystal structure, similar electronegativity and valency.
Phase — physically homogeneous, distinct portion of a system.

FormulaPhase rule and lever rule

Gibbs phase rule: $F = C - P + 2$ (for condensed metallurgical systems at constant pressure: $F = C - P + 1$ ) $F$ = degrees of freedom; $C$ = components; $P$ = phases.

Lever rule (two-phase region, overall composition $C_{0}$ , phase compositions $C_{α}$ and $C_{L}$ ):

W_{L} = \frac{C _{0} - C _{α}}{C _{L} - C _{α}}, W_{α} = \frac{C _{L} - C _{0}}{C _{L} - C _{α}}

Invariant reactions

Reaction	Type
Eutectic	Liquid → Solid α + Solid β (on cooling)
Eutectoid	Solid γ → Solid α + Solid β
Peritectic	Liquid + Solid α → Solid β

Iron–carbon (Fe–Fe₃C) diagram — key points

Steels — up to about 2.1% C (commonly quoted as 2%); cast irons — above (practically 2–4% C).
Eutectoid point: about 0.76–0.8% C at 727 °C — austenite → pearlite (lamellar ferrite + cementite).
- Hypoeutectoid steels (< 0.8% C): ferrite + pearlite (e.g. mild steel).
- Hypereutectoid steels (> 0.8% C): pearlite + cementite.
Eutectic point: about 4.3% C at 1147 °C — liquid → ledeburite (austenite + cementite).
Phases/microconstituents: ferrite (α) — soft, ductile BCC; austenite (γ) — FCC, non-magnetic, stable at high temperature; cementite (Fe₃C) — hard, brittle (6.67% C); pearlite; martensite (non-equilibrium, formed by rapid quenching — very hard); bainite.

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