Brillouin zone

Remarks

The translates $B + G$, $G \in \Lambda^*$, cover $\R^n$ and overlap only on their boundaries. So $B$ is a fundamental domain for $\Lambda^*$, and its volume is $(2\pi)^n / V$, where $V$ is the volume of a primitive cell of $\Lambda$. Identifying opposite faces, $B$ becomes the torus $\R^n / \Lambda^*$.

Example: for $\Lambda = a\Z$ with $a > 0$, $\Lambda^* = (2\pi/a)\Z$ and $B = [-\pi/a, \pi/a]$. Check: $\abs{k} \le \abs{k - 2\pi m/a}$ for all integers $m$ holds exactly when $\abs{k} \le \pi/a$.

Non-example: for $\Lambda = a\Z^2$ the square $[0, 2\pi/a]^2$ is a fundamental domain of $\Lambda^*$, but it is not the Brillouin zone. The point $k = (1.9\pi/a, 0)$ lies in it, yet it is closer to $G = (2\pi/a, 0)$ than to $0$. The Brillouin zone of $a\Z^2$ is $[-\pi/a, \pi/a]^2$.