Bravais Lattices, Bases, and Crystal Structures

A crystal is periodic: some group of atoms repeats at every point of an infinite regular array. The previous lesson took the cubic unit cell for granted and summed the Coulomb energy over it. Here the array itself is the object of study. Two ingredients build every crystal: an abstract lattice of translation-equivalent points, and a group of atoms — the basis — attached to each point. Separating them makes the symmetry classification tractable and gives the vocabulary that diffraction and band theory both use.

The Bravais lattice

The two definitions agree: translating the lattice by any $R$ maps it onto itself, so every point has identical surroundings. The primitive vectors are not unique — any set that generates the same point set will do — but their number equals the dimension, and the parallelepiped they span, the primitive cell, contains exactly one lattice point.

A physical crystal rarely has a single atom at each lattice point. The full structure is a lattice plus a basis: a fixed set of atoms, with positions $d_{1}, \dots, d_{p}$ relative to a lattice point, repeated at every $R$ .

crystal structure = Bravais lattice + basis .

The distinction is sharp because many common lattices are not Bravais. The two-dimensional honeycomb (the graphene sheet) is the standard example: no single set of primitive vectors reaches every vertex, because neighboring vertices point in opposite directions and are not translation-equivalent. The honeycomb is a triangular Bravais lattice with a two-atom basis.

The honeycomb is not a Bravais lattice. It is a triangular Bravais lattice (open circles, primitive vectors a1 and a2) with a two-atom basis, one A atom and one B atom (filled), attached to every lattice point.

Primitive, conventional, and Wigner-Seitz cells

A primitive cell is any region that fills space with no overlap under the lattice translations and contains exactly one lattice point. Its volume is fixed regardless of shape,

v_{c} = ∣ a_{1} \cdot (a_{2} \times a_{3})∣,

because a smaller region could not tile space and a larger one would hold more than one point. The primitive cell of the fcc lattice is a rhombohedron spanned by vectors to three face centers; it is oblique and awkward, which is why the conventional cell — the full cube with four lattice points — is used instead. A conventional cell is a larger, more symmetric unit that displays the cubic symmetry at the cost of holding several lattice points.

The one primitive cell with the full point symmetry of the lattice is the Wigner-Seitz cell: the region of space closer to a given lattice point than to any other. It is constructed by drawing the vectors from one point to its neighbors, bisecting each with a perpendicular plane, and taking the smallest enclosed volume.

Wigner-Seitz construction in two dimensions. Bisect every bond from the central point to its neighbors with a perpendicular line; the smallest enclosed polygon (shaded) is the Wigner-Seitz cell, a primitive cell centered on the point with the full symmetry of the lattice.

The seven crystal systems and fourteen Bravais lattices

The point-symmetry operations that leave a Bravais lattice invariant (rotations, reflections, inversion) fall into seven distinct groups, the crystal systems. Allowing the lattice to be centered — an extra point at the body center, face centers, or one pair of opposite faces — without lowering the point symmetry generates additional lattices within a system. The full count is fourteen Bravais lattices, first enumerated by Bravais.

The fourteen Bravais lattices arranged by crystal system (rows) and centering type (columns): P primitive, C base-centered, I body-centered, F face-centered. A cell icon appears where that combination is a distinct lattice; the trigonal R and hexagonal cells sit in the primitive column.

The seven systems and their cell parameters, with the centering types each admits, are collected below.

System	Axes	Angles	Bravais lattices
Triclinic	$a \neq = b \neq = c$	all $\neq = 90°$	P
Monoclinic	$a \neq = b \neq = c$	two $= 90°$	P, C
Orthorhombic	$a \neq = b \neq = c$	all $= 90°$	P, C, I, F
Tetragonal	$a = b \neq = c$	all $= 90°$	P, I
Trigonal	$a = b = c$	all $= α \neq = 90°$	R
Hexagonal	$a = b \neq = c$	$90°, 90°, 120°$	P
Cubic	$a = b = c$	all $= 90°$	P, I, F

Some centerings are absent because they duplicate a smaller cell of the same or higher symmetry. A face-centered tetragonal cell, for instance, is a body-centered tetragonal cell with smaller axes, so it is not counted separately.

Point groups and space groups

Two further classifications refine the fourteen lattices to real crystals. The point group of a crystal is the set of symmetry operations — proper and improper rotations — that fix at least one point and map the structure onto itself. The crystallographic restriction (only 1-, 2-, 3-, 4-, and 6-fold axes are compatible with translational periodicity) limits these to 32 crystallographic point groups. Adding the translations, including screw axes (rotation plus a fractional translation) and glide planes (reflection plus a fractional translation), gives the 230 space groups that exhaust the ways a crystal can be symmetric in three dimensions. The point group governs macroscopic tensor properties — piezoelectricity, birefringence — while the space group is what a diffraction experiment ultimately determines.

Miller indices

Directions and planes in a crystal are labeled relative to the axes $a_{1}, a_{2}, a_{3}$ .

For a plane intercepting the axes at $(2, 1, 1)$ in lattice units, the reciprocals are $(\frac{1}{2}, 1, 1)$ , which clear to $(122)$ . The prescription looks arbitrary until the reciprocal lattice makes it natural: $(hk l)$ is the plane perpendicular to the reciprocal-lattice vector $h b_{1} + k b_{2} + l b_{3}$ . In a cubic crystal $[hk l]$ is normal to $(hk l)$ , and the spacing between adjacent $(hk l)$ planes is

d_{hk l} = \frac{a}{h ^{2} + k ^{2} + l ^{2}} .

The three low-index planes of a cubic crystal. (100) cuts one axis, (110) cuts two, (111) cuts all three; each shaded plane is the member of its family nearest the origin.

Close packing and the important structures

Stacking equal spheres to fill space efficiently gives two arrangements, both with packing fraction $0.740$ : face-centered cubic, with layer sequence $A B C A B C$ , and hexagonal close-packed (hcp), with sequence $A B A B$ . The hcp structure is a hexagonal Bravais lattice with a two-atom basis; its ideal axial ratio is

\frac{c}{a} = \frac{8}{3} = 1.633.

The packing fraction of a structure is the volume of the atoms in a conventional cell divided by the cell volume, with atoms taken as touching spheres.

The remaining important structures are decorated cubic lattices:

NaCl (rock salt): fcc lattice, two-atom basis (a cation at the origin, an anion at the cube center of the fcc cell); six-fold coordination.
CsCl: simple-cubic lattice with a two-atom basis (one ion at the corner, one at the body center); eight-fold coordination. It is not body-centered cubic, because the two ions differ.
Diamond: two interpenetrating fcc lattices offset by $\frac{1}{4} (a_{1} + a_{2} + a_{3})$ ; each atom is tetrahedrally coordinated to four neighbors. Packing fraction $3 π /16 = 0.340$ , low because the covalent bonding fixes the tetrahedral angle rather than close packing.
Zincblende: the diamond structure with the two sublattices occupied by different elements (ZnS, GaAs); the loss of inversion symmetry is what makes these crystals piezoelectric and optically active.
Wurtzite: the hexagonal analog of zincblende, based on hcp stacking (ZnO, GaN).

The diamond structure as two interpenetrating fcc lattices. The second sublattice (open) is shifted by one quarter of the body diagonal from the first (filled); each atom bonds tetrahedrally to four neighbors on the other sublattice.

The catalog of structures reduces to a small number of lattices and bases, and the symmetry that classifies them is the same symmetry that will restrict the allowed diffraction spots. Reading a diffraction pattern requires the dual description of the lattice in momentum space: the reciprocal lattice.