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Crystalline solids consist of atoms, ions, or molecules arranged in a periodically repeating three-dimensional pattern. The smallest repeating unit that captures the full symmetry and composition of the crystal is the unit cell. There are 14 Bravais lattices in three dimensions, grouped into 7 crystal systems (cubic, tetragonal, orthorhombic, hexagonal, trigonal, monoclinic, triclinic). Common structure types — FCC, BCC, HCP for metals; rock salt, fluorite, zinc blende for ionic compounds — arise from different ways of packing spheres and filling interstitial sites. The choice of unit cell determines how you calculate density, coordination numbers, and stoichiometry from crystallographic data.
Crystals are the most ordered state of matter. Unlike liquids or glasses, where atomic positions are random or only locally organized, a crystal has long-range order — if you know the position and identity of every atom in one small region, you can predict the contents of the entire solid by applying translation operations. The unit cell is the fundamental building block of this translational symmetry: the smallest parallelepiped that, when repeated in all three directions, reproduces the full crystal.
The geometry of the unit cell is defined by six parameters: three edge lengths (a, b, c) and three angles (alpha, beta, gamma). These parameters, combined with the lattice type, determine the crystal system. In the cubic system, a = b = c and all angles are 90 degrees, but three distinct lattice types exist: primitive (P), body-centered (I), and face-centered (F). The distinction matters because different lattice types pack atoms differently — FCC achieves 74% packing efficiency (the theoretical maximum for equal spheres), while BCC reaches only 68%. These packing differences directly determine properties like density, ductility, and slip systems in metals.
For ionic compounds, the structure depends not just on packing but on the radius ratio of cation to anion. Large cations relative to anions favor high coordination numbers (8, as in CsCl); intermediate ratios favor octahedral coordination (6, as in NaCl); small cations favor tetrahedral coordination (4, as in ZnS). This radius ratio rule is approximate — it ignores covalent character, polarization, and entropy — but it correctly predicts the structure of most simple ionic compounds and provides the starting framework for understanding more complex structures.
Counting atoms within a unit cell requires careful bookkeeping because atoms on corners, edges, and faces are shared with neighboring cells. A corner atom contributes 1/8, an edge atom 1/4, a face atom 1/2, and a body-center atom 1. This counting directly gives you stoichiometry (the ratio of different atoms in the formula unit) and enables density calculations from diffraction data. The connection between unit cell parameters, atom positions, and macroscopic properties like density is one of the most practically useful results in materials chemistry — it allows you to go from an X-ray diffraction pattern to a complete structural model of a new material.