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A crystal is a solid whose atoms are arranged in a periodically repeating pattern. This periodicity is captured by a Bravais lattice — the set of all points R = n_1 a_1 + n_2 a_2 + n_3 a_3, where a_i are primitive lattice vectors and n_i are integers. In three dimensions there are exactly 14 distinct Bravais lattices grouped into 7 crystal systems (cubic, tetragonal, orthorhombic, hexagonal, trigonal, monoclinic, triclinic), each distinguished by the symmetry operations that leave the lattice invariant. The physical crystal is described by placing a basis (one or more atoms) at each lattice point.
Condensed matter physics begins with a question that sounds deceptively simple: how are the atoms in a solid arranged? For crystalline solids — which include most metals, semiconductors, and many ceramics — the answer is a periodic arrangement that repeats identically throughout space. The mathematical abstraction of this periodicity is the Bravais lattice: an infinite set of discrete points generated by R = n_1 a_1 + n_2 a_2 + n_3 a_3, where the three primitive vectors a_i define the lattice and n_i range over all integers. The defining property is that the lattice looks exactly the same from every lattice point — every point has an identical environment.
In three dimensions, the constraints of symmetry allow exactly 14 distinct Bravais lattices, organized into 7 crystal systems. The crystal systems are defined by the relationships among the lattice parameters (edge lengths a, b, c and angles alpha, beta, gamma): cubic has a = b = c with all right angles, hexagonal has a = b with gamma = 120 degrees, and so on down to triclinic with no constraints at all. Within each system, you can place additional lattice points at the body center, face centers, or base centers — but many of these centerings turn out to be equivalent to a lattice in a different (lower-symmetry) system after redefining the primitive vectors. Eliminating all redundancies leaves exactly 14.
A real crystal is more than just a lattice — it is a lattice plus a basis, the set of atoms placed at each lattice point. Monatomic metals like copper have a one-atom basis on an FCC lattice. Sodium chloride has a two-atom basis (Na and Cl) on an FCC lattice. Diamond and silicon have a two-atom basis on FCC where both atoms are the same element but sit at inequivalent positions. The distinction between lattice and basis is critical: the lattice captures translational symmetry, while the basis captures what sits at each point. Two completely different materials can share the same Bravais lattice but differ in their basis.
The full symmetry of a crystal includes not just translations but also rotations, reflections, and inversions that map the crystal onto itself. These additional symmetries define the point group (symmetry operations that leave at least one point fixed) and the space group (the full set of symmetry operations including translations, screw axes, and glide planes). There are 32 crystallographic point groups and 230 space groups. While you rarely need all 230 in a physics course, the key insight is that symmetry constrains everything — the allowed vibrational modes, electronic band structure, optical properties, and response to external fields are all dictated by the space group. Understanding the lattice is the first step toward understanding the solid.