Crystal Geometry — Math at the Atomic Scale
Crystal Geometry
Every metal, ceramic, and semiconductor is made of atoms arranged in precise geometric patterns. The math of these patterns determines everything: strength, conductivity, transparency, and more.
Unit Cells — Geometry Repeated Billions of Times
Atoms in a crystal arrange themselves in repeating 3D patterns called unit cells. The three most common are:
- BCC (Body-Centered Cubic): atoms at corners + 1 in center → iron, chromium, tungsten
- FCC (Face-Centered Cubic): atoms at corners + 1 on each face → aluminum, copper, gold
- HCP (Hexagonal Close-Packed): hexagonal layers → titanium, zinc, magnesium
Geometry Determines Density
The atomic packing factor (APF) — how much of the cube is filled — is pure geometry:
For FCC:
The face diagonal of the cube equals $4r$ (four atomic radii). By the Pythagorean theorem:
$$a\sqrt{2} = 4r \quad \Rightarrow \quad a = 2r\sqrt{2}$$
The FCC unit cell contains 4 atoms (corners share 1/8, faces share 1/2):
$$\text{APF}_{\text{FCC}} = \frac{4 \times \frac{4}{3}\pi r^3}{(2r\sqrt{2})^3} = \frac{\frac{16}{3}\pi r^3}{16\sqrt{2}\, r^3} = \frac{\pi}{3\sqrt{2}} \approx 0.74$$
This means 74% of the space is filled — the theoretical maximum for identical spheres! This was conjectured by Kepler in 1611 and only proven mathematically in 1998.
Copper has an FCC structure with atomic radius $r = 128$ pm. Calculate the lattice parameter and theoretical density. (Atomic mass of Cu = 63.55 g/mol, $N_A = 6.022 \times 10^{23}$)
Lattice parameter:
$$a = 2r\sqrt{2} = 2(128 \times 10^{-12})\sqrt{2} = 3.62 \times 10^{-10} \text{ m}$$
Density: (4 atoms per FCC cell)
$$\rho = \frac{n \times M}{N_A \times a^3} = \frac{4 \times 63.55}{6.022 \times 10^{23} \times (3.62 \times 10^{-10})^3}$$
$$= \frac{254.2}{6.022 \times 10^{23} \times 4.75 \times 10^{-29}} = \frac{254.2}{28.6 \times 10^{-6}} \approx 8{,}890 \text{ kg/m}^3$$
The measured density of copper is 8,960 kg/m³ — our calculation is within 1% using nothing but geometry, fractions, and exponents!
The Math You Need
- Pythagorean theorem: relates atomic radius to cell size
- Fractions: counting shared atoms (1/8 corners, 1/2 faces)
- Volume formulas: spheres and cubes
- Scientific notation: working at the $10^{-10}$ meter scale
Geometry and fractions at the atomic scale determine whether a material is strong or weak, conducts electricity or doesn't. The basic math you're learning applies at every scale of the universe.