Reciprocal lattice

Remarks

In dimension $3$, $b_1 = 2\pi \dfrac{a_2 \times a_3}{a_1 \cdot (a_2 \times a_3)}$, and $b_2, b_3$ follow by cyclic permutation. The reciprocal of $\Lambda^*$ is $\Lambda$, and the primitive cells have volumes related by $V^* = (2\pi)^n / V$. Mathematicians often use the dual lattice $\set{k : k \cdot R \in \Z}$, which is $\Lambda^*$ divided by $2\pi$.

Example: for $\Lambda = a\Z^n$ with $a > 0$, $\Lambda^* = (2\pi/a)\Z^n$. Indeed $k \cdot (a e_j) = a k_j \in 2\pi\Z$ for every $j$ holds exactly when each $k_j$ is a multiple of $2\pi/a$.

Non-example: for $\Lambda = \Z^2$, the vector $k = (\pi, 0)$ is not in $\Lambda^*$, because $k \cdot (1, 0) = \pi \notin 2\pi\Z$. The wave $e^{i\pi x_1}$ changes sign under the shift by $(1, 0)$.