The Reciprocal Lattice and Brillouin Zones

Any function with the periodicity of a crystal — the electron density, the potential, the charge distribution — is a sum of plane waves, and only certain wavevectors appear in that sum. Those allowed wavevectors form a lattice of their own, the reciprocal lattice. It is the natural arena for diffraction, for Bloch's theorem, and for the whole of band theory: a wave scatters off a crystal only when its wavevector change is a reciprocal-lattice vector, and an electron's crystal momentum is defined only up to one. This lesson constructs the reciprocal lattice from the direct (Bravais) lattice and partitions reciprocal space into Brillouin zones.

Definition and construction

Let a crystal property have the periodicity of a Bravais lattice with primitive vectors $a_{1}, a_{2}, a_{3}$ . Expand it in plane waves $e^{i k \cdot r}$ . Periodicity $f (r + R) = f (r)$ for every lattice vector $R$ requires $e^{i k \cdot R} = 1$ , i.e. $k \cdot R = 2 π \times integer$ . The set of $k$ satisfying this for all $R$ is the reciprocal lattice.

The generating vectors follow explicitly. With $v_{c} = a_{1} \cdot (a_{2} \times a_{3})$ the primitive cell volume,

b_{1} = 2 π \frac{a _{2} \times a _{3}}{v _{c}}, b_{2} = 2 π \frac{a _{3} \times a _{1}}{v _{c}}, b_{3} = 2 π \frac{a _{1} \times a _{2}}{v _{c}} .

Each $b_{i}$ is perpendicular to two of the direct primitive vectors, and the $2 π$ normalization gives $b_{i} \cdot a_{j} = 2 π δ_{ij}$ directly. A general reciprocal vector is $G = h b_{1} + k b_{2} + l b_{3}$ with integer $h, k, l$ ; then $G \cdot R = 2 π (h n_{1} + k n_{2} + l n_{3})$ is $2 π$ times an integer, as required.

Three properties follow immediately:

The reciprocal of the reciprocal lattice is the original direct lattice.
The reciprocal-lattice primitive cell has volume $(2 π)^{3} / v_{c}$ .
The units of $G$ are inverse length; the reciprocal lattice lives in wavevector (momentum) space.

A direct lattice (left) and its reciprocal (right) in two dimensions. The oblique direct cell with primitive vectors a1, a2 maps to a reciprocal cell with b1 perpendicular to a2 and b2 perpendicular to a1; a denser direct lattice gives a sparser reciprocal one.

Reciprocal lattices of the cubic families

For a simple-cubic lattice with $a_{i} = a \overset{e}{^}_{i}$ , the reciprocal vectors are $b_{i} = (2 π / a) \overset{e}{^}_{i}$ : another simple-cubic lattice with constant $2 π / a$ .

The centered cubics swap. Applying the cross-product formula to the fcc primitive vectors $a_{1} = \frac{a}{2} (\overset{y}{^} + \overset{z}{^})$ , $\vec a_2 = \tfrac{a}{2}(\hat z

\hat x) $,$ \vec a_3 = \tfrac{a}{2}(\hat x + \hat y) $g i v esr ec i p r oc a l v ec t or so f t h e f or m$ (2\pi/a)(\hat y + \hat z - \hat x) $, w hi c h g e n er a t e a * * b o d y - ce n t er e d c u bi c * * l a tt i ce w i t h c u bi cco n s t an t$ 4\pi/a$. Conversely the reciprocal of bcc is fcc. This duality controls diffraction: X-rays scattered from an fcc crystal produce spots on a bcc reciprocal lattice.

Direct lattice	Reciprocal lattice
Simple cubic, constant $a$	Simple cubic, constant $2 π / a$
Body-centered cubic	Face-centered cubic
Face-centered cubic	Body-centered cubic

Reciprocal vectors and lattice planes

The reason Miller indices use reciprocals is that each reciprocal-lattice vector is tied to a family of direct-lattice planes.

A reciprocal-lattice vector G is normal to its family of direct lattice planes, and its length fixes the plane spacing through d = 2 pi over the magnitude of G. Denser planes (larger index) mean a longer G.

The first Brillouin zone

The Wigner-Seitz cell of the reciprocal lattice is the first Brillouin zone. It is built exactly as in the direct lattice: draw vectors from a reciprocal point to its neighbors, bisect each with a perpendicular plane, and take the enclosed volume.

Because the reciprocal of fcc is bcc, the first Brillouin zone of an fcc crystal is the Wigner-Seitz cell of the bcc lattice: a truncated octahedron with six square faces (along the cube axes) and eight hexagonal faces (along the body diagonals). Its high-symmetry points carry standard labels used throughout band theory: $Γ$ at the center, $X$ at a square face center, $L$ at a hexagonal face center, and $K$ on an edge.

The first Brillouin zone of the fcc lattice is a truncated octahedron: six square faces along the cube axes, eight hexagonal faces along the body diagonals. Gamma is the zone center, X a square-face center, L a hexagon-face center, K a hexagon edge midpoint.

Higher zones

The perpendicular bisectors of all reciprocal vectors — the Bragg planes — carve reciprocal space into concentric regions. The first zone is the innermost cell; the second zone is the set of fragments lying between the first and second Bragg planes; the third lies beyond those. Every zone has the same total volume, $(2 π)^{3} / v_{c}$ , since each is a full primitive cell reassembled from pieces. Translating the fragments of the $n$ -th zone by suitable reciprocal vectors folds them back into the first zone, which is why band structures can always be drawn in the reduced-zone scheme.

First, second, and third Brillouin zones of a two-dimensional square lattice. Zone 1 is the central square (nearest Bragg planes); zone 2 is the four triangles beyond it; zone 3 is the outer fragments. Each has equal area and folds back onto zone 1 by a reciprocal-lattice translation.

The reciprocal lattice and its first zone are the coordinate system for everything that follows. Diffraction spots sit at reciprocal-lattice points, and the Bragg and Laue conditions are two statements of the same requirement, that the scattering vector be a reciprocal-lattice vector.