Crystal systems and Bravais lattices
Every crystalline material's translational symmetry falls into one of 7 systems and 14 Bravais lattices. Combined with point symmetry that yields 32 crystallographic point groups and, once screw axes and glide planes are added, the 230 space groups.
| System | Axes | Angles | Bravais lattices | Point groups | Elements |
|---|---|---|---|---|---|
| Cubic (isometric) | a = b = c | α = β = γ = 90° | Simple (P), Body-centered (I), Face-centered (F) | 5 | 38 |
| Tetragonal | a = b ≠ c | α = β = γ = 90° | Simple (P), Body-centered (I) | 7 | 3 |
| Orthorhombic | a ≠ b ≠ c | α = β = γ = 90° | Simple (P), Base-centered (C), Body-centered (I), Face-centered (F) | 3 | 7 |
| Hexagonal | a = b ≠ c | α = β = 90°, γ = 120° | Simple (P) | 7 | 33 |
| Trigonal (rhombohedral) | a = b = c (rhombohedral setting) | α = β = γ ≠ 90° | Rhombohedral (R) | 5 | 6 |
| Monoclinic | a ≠ b ≠ c | α = γ = 90°, β ≠ 90° | Simple (P), Base-centered (C) | 3 | 5 |
| Triclinic | a ≠ b ≠ c | α ≠ β ≠ γ ≠ 90° | Simple (P) | 2 | 1 |
Cubic (isometric)
- Axes
- a = b = c
- Angles
- α = β = γ = 90°
- Lattices
- 3
Cu, Al, Fe-α (BCC), NaCl, diamond, spinel MgAl₂O₄
Tetragonal
- Axes
- a = b ≠ c
- Angles
- α = β = γ = 90°
- Lattices
- 2
β-Sn, rutile TiO₂, zircon, martensite (BCT)
Orthorhombic
- Axes
- a ≠ b ≠ c
- Angles
- α = β = γ = 90°
- Lattices
- 4
α-S, cementite Fe₃C, aragonite CaCO₃, olivine
Hexagonal
- Axes
- a = b ≠ c
- Angles
- α = β = 90°, γ = 120°
- Lattices
- 1
Mg, Zn, Ti-α, graphite, ZnO (wurtzite), ice Ih
Trigonal (rhombohedral)
- Axes
- a = b = c (rhombohedral setting)
- Angles
- α = β = γ ≠ 90°
- Lattices
- 1
α-quartz, calcite CaCO₃, Bi, corundum Al₂O₃
Monoclinic
- Axes
- a ≠ b ≠ c
- Angles
- α = γ = 90°, β ≠ 90°
- Lattices
- 2
β-S, gypsum, monoclinic ZrO₂, augite
Triclinic
- Axes
- a ≠ b ≠ c
- Angles
- α ≠ β ≠ γ ≠ 90°
- Lattices
- 1
Kaolinite, albite, CuSO₄·5H₂O
Why crystallography earns its place in an engineering workflow
14 Bravais lattices
Every crystalline material's translational symmetry is described by one of 14 Bravais lattices — the 7 crystal systems combined with permitted centerings (P, I, F, C, R). Adding point-symmetry operations yields 32 crystallographic point groups and, with translations (screw axes, glide planes), 230 space groups.
Common metallic structures
FCC (Cu, Al, Ni, austenite): 12 slip systems, ductile, close-packed (APF 0.74). BCC (Fe-α, W, Mo): stronger but with ductile-brittle transition. HCP (Mg, Ti-α, Zn): limited slip systems → anisotropy and lower formability.
Structure–property links
Slip systems govern ductility; packing density influences density and diffusion; symmetry controls anisotropy of stiffness, thermal expansion and piezoelectricity (only non-centrosymmetric groups can be piezoelectric). Polymorphism (e.g. Fe BCC↔FCC, ZrO₂ monoclinic↔tetragonal) underpins heat treatment and transformation toughening.
Defects
Point defects (vacancies, interstitials, dopants) drive diffusion and conductivity; dislocations carry plasticity and are obstructed by solutes, precipitates and grain boundaries (strengthening mechanisms); planar defects (grain boundaries, stacking faults, twins) and volume defects (pores, inclusions) control strength, toughness and failure.
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