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Construct Bragg planes by taking the midpoints between the reciprocal origin and the other reciprocal lattice points. In one dimension these are points at $k=m\pi /a$, for nonzero integer $m$.

The first Brillouin zone is the interval bounded by the nearest two such points:The second Brillouin zone consists of the next intervals on both sides:Boundary points may be assigned to one adjacent zone to avoid double counting. The first zone is reached from the origin without crossing a Bragg point; the second is reached after crossing one. Each zone has total length $2\pi /a$. In particular, the two separated pieces of the second zone are not the entire translated first-zone interval centered at $2\pi /a$.

A Wigner-Seitz cell is the region of real space at least as close to a chosen Bravais lattice point as to any other. It is bounded by perpendicular bisectors of vectors to neighboring lattice points and is a primitive unit cell.

A reciprocal lattice vector $\mathbf{G}$ satisfies $e^{i\mathbf{G}\cdot \mathbf{R}}=1$ for every lattice translation $\mathbf{R}$. With primitive vectors ${\mathbf{a}}_i$, reciprocal primitive vectors satisfy ${\mathbf{a}}_i\cdot {\mathbf{b}}_j=2\pi {\delta }_{ij}$, and $\mathbf{G}={\sum }_j{n}_j{\mathbf{b}}_j$ for integers $n_j$.

The first Brillouin zone is the Wigner-Seitz cell of that reciprocal lattice, centered at $\mathbf{k}=0$. Its boundaries are Bragg planes, $\mathbf{k}\cdot \mathbf{G}=G^2/2$. Higher Brillouin zones are the regions separated from the origin by successively more such planes: an interior point of zone $j$ has crossed exactly $j-1$ planes. A second Brillouin zone is generally disconnected in this extended-zone representation; it is not just a translated first-zone rectangle.

For the first two Brillouin zones of a rectangular lattice, write $K=\pi /a$ and $L=\pi /b$, so $0<K<L$ and the reciprocal lattice vectors are $(2nK,2mL)$. The perpendicular bisectors to the four nearest axial reciprocal points encloseThe second Brillouin zone consists of points beyond exactly one Bragg plane. In the first quadrant, it is the union of the following two regions:Reflect these regions in both axes. The first is a trapezoidal side lobe; the second is half of a triangular top lobe. The sloping boundary is the bisector for $\mathbf{G}=(2K,2L)$. On the side lobe, $k_x<2K$ is the bisector condition for $\mathbf{G}=(4K,0)$: crossing that plane too would enter the third or a higher zone. These additional bisectors are why four rectangles pasted onto the first zone would be incorrect.

Equivalently, the full right lobe has verticesand the top lobe has vertices $(-K,L),(K,L),(0,L+K^2/L)$. Add their left and bottom reflections. Their total area isequal to the first-zone area, as required by reciprocal-space state counting. These formulas remain valid for every $a>b$, including very elongated rectangles; boundaries themselves can be shared by convention.