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Solids and Crystal Structures

In a solid, atoms, ions, or molecules lock into fixed positions. In a crystal they lock into a regular, repeating pattern — and that pattern decides the solid's density, strength, and melting point.

Unit Cells: Simple Cubic, BCC, and FCC

A crystal is built by repeating one small box, the unit cell. Atoms at the corners are shared between eight neighbouring cells, so each corner atom counts as 1/8 of an atom in the cell; an atom in a face centre is shared by two cells and counts as 1/2.

Simple cubic (SC): atoms only at the corners. Atoms per cell: $8 \times \tfrac{1}{8} = 1$. Coordination number (nearest neighbours): 6. Packing efficiency: 52%.
Body-centred cubic (BCC): corner atoms plus one atom in the centre of the cell. Atoms per cell: $1 + 1 = 2$. Coordination number: 8. Packing efficiency: 68%.
Face-centred cubic (FCC): corner atoms plus one atom in the centre of each face. Atoms per cell: $8 \times \tfrac{1}{8} + 6 \times \tfrac{1}{2} = 4$. Coordination number: 12. Packing efficiency: 74%.

Packing efficiency is the fraction of the cell's volume actually filled by the atoms. FCC (and hexagonal close packing) reaches 74%, the most efficient way spheres can pack — that is why most metals crystallize as FCC or hexagonal close-packed.

Example: The FCC packing efficiency. In FCC, atoms touch along the face diagonal, so the face diagonal is $4r$ while the edge length is $a$: $a\sqrt{2} = 4r$, so $a = 2\sqrt{2}\,r$. Four atoms per cell occupy $4 \times \tfrac{4}{3}\pi r^3$ of a cell of volume $a^3 = (2\sqrt{2}\,r)^3 = 16\sqrt{2}\,r^3$: $$\text{Efficiency} = \frac{4 \times \frac{4}{3}\pi r^3}{16\sqrt{2}\,r^3} = \frac{\pi}{3\sqrt{2}} = 0.740$$ 74% of the volume is filled; 26% is empty space.

Density from a Unit Cell

If you know the edge length $a$, the atoms per cell, and the molar mass, you can calculate the density: mass of one cell divided by its volume.

Example: Copper is FCC with edge length $a = 361$ pm and molar mass 63.55 g/mol. Cell volume: $V = a^3 = (3.61 \times 10^{-8}\ \text{cm})^3 = 4.70 \times 10^{-23}\ \text{cm}^3$. Mass of one cell (4 atoms): $$m = \frac{4 \times 63.55\ \text{g/mol}}{6.022 \times 10^{23}\ \text{atoms/mol}} = 4.22 \times 10^{-22}\ \text{g}$$ $$\text{Density} = \frac{4.22 \times 10^{-22}\ \text{g}}{4.70 \times 10^{-23}\ \text{cm}^3} = 8.97\ \text{g/cm}^3$$ This matches the accepted density of copper, 8.96 g/cm³ (the small difference is rounding in the edge length).

Types of Solids

Metallic: metal cations in a sea of delocalized electrons — e.g., Cu, Fe, Al. Conduct electricity and heat, malleable and ductile, usually high melting points.
Ionic: cations and anions held by electrostatic attraction — e.g., NaCl, MgO. Hard and brittle, high melting points, conduct only when molten or dissolved.
Covalent network: atoms joined by a continuous network of covalent bonds — e.g., diamond, quartz (SiO₂), graphite. Extremely hard, very high melting points; graphite conducts along its layers.
Molecular: discrete molecules held by intermolecular forces — e.g., ice (H₂O), dry ice (CO₂), I₂. Soft, low melting points, poor conductors.

Phase Diagrams

A phase diagram maps which phase (solid, liquid, gas) is stable at each combination of temperature and pressure. Two points matter most:

Triple point: the one temperature and pressure where solid, liquid, and gas all coexist in equilibrium.
Critical point: above this temperature and pressure, the liquid–gas boundary vanishes — the substance becomes a supercritical fluid, with no distinct liquid or gas phase.

For carbon dioxide, the triple point is 5.11 atm and −56.6 °C, and the critical point is 73 atm and 31.1 °C. Because the triple-point pressure is above 1 atm, liquid CO₂ cannot exist at ordinary pressure — dry ice sublimes directly to gas at −78.5 °C. For water, the critical point is 218 atm and 374 °C, and its triple point is 0.006 atm (4.58 torr) and 0.01 °C. Water is unusual in another way: the solid–liquid boundary slopes backwards, so ice is less dense than liquid water — which is why ice floats.

Practice Problems

Polonium is simple cubic with edge length 335 pm. What is the atomic radius of polonium? Calculate its density (M = 209 g/mol).
Iron is BCC with edge length 287 pm. Calculate its density (M = 55.85 g/mol).
How many atoms are in an FCC unit cell, and what is the coordination number of each atom?
Explain why dry ice (solid CO₂) sublimes at 1 atm instead of melting, using CO₂'s triple point of 5.11 atm.