Reciprocal lattices and Brillouin zones

Define reciprocal primitive vectors by $\mathbf a_i\cdot\mathbf b_j=2\pi\delta_{ij}$. An integer combination G of the b vectors then satisfies $e^{i\mathbf G\cdot\mathbf R}=1$ for every direct-lattice translation R. The dot product is dimensionless, with a in length units and b in inverse-length units. Some crystallographic conventions omit two pi and put it into the Fourier exponential instead. Both are legitimate, but combining a no-two-pi reciprocal vector with our exponential produces incorrect diffraction angles and zone widths.

In three dimensions a reciprocal vector is obtained from a cross product divided by the oriented primitive volume. Its direction is perpendicular to the other two direct vectors, not generally parallel to its matching direct vector. Orthogonal rectangular lattices make the construction look simpler than it is. For a sheared cell the reciprocal grid shears oppositely so the dual scalar products remain fixed. Computing those products explicitly is an effective check on a proposed reciprocal basis, including signs and factors of two pi.

The first Brillouin zone is the Wigner–Seitz cell around the reciprocal origin. Its boundary points are equally distant from the origin and a neighboring reciprocal point. Squaring that distance condition gives a plane $\mathbf q\cdot\mathbf G=|\mathbf G|^2/2$. Intersecting the relevant half-spaces constructs the zone. For a rectangular lattice the boundaries are simple coordinate limits; for an oblique lattice the shortest reciprocal vectors can produce a polygon whose faces are not obtained by taking half of each basis vector independently.

Wavevectors differing by a reciprocal vector give identical phase factors on the Bravais sites. Choosing one representative in the first zone removes that redundancy for lattice amplitudes. A multi-atom basis can acquire corresponding internal phase conventions, so eigenvectors may need to transform when a wavevector is folded. The physical displacement pattern remains equivalent when these conventions are handled consistently. Folding does not mean that a freely propagating probe in empty space has literally changed its wavevector; it is a representation of periodic-medium states.

For a periodic sample with N primitive cells, there are N allowed wavevectors in one Brillouin zone per branch, ignoring the bookkeeping of equivalent boundary points. The allowed spacing follows from the finite sample's periodic boundary condition. Increasing real-space cell volume by an integer supercell factor shrinks the zone by that factor and increases the number of folded branches. The total count of states or vibrational coordinates is unchanged. Counting both folded branches and the original full zone would therefore double-count physical modes.

Reciprocal geometry alone does not specify an energy or frequency dispersion. The zone tells us the domain of independent wavevector labels, while forces and masses determine phonon frequencies. Similarly, the reciprocal lattice gives candidate diffraction locations, but basis interference can remove their intensity. Separating these roles avoids drawing a band gap simply because a zone boundary exists or assuming every reciprocal point produces an observable peak. Geometry constrains the calculation; it does not replace the dynamical or scattering-amplitude information.