 Solution

The reciprocal lattice is the set of wavevectors $\mathbf{G}$ satisfying $e^{i\mathbf{G}\cdot \mathbf{R}}=1$ for every direct Bravais lattice translation $\mathbf{R}$. Equivalently, $\mathbf{G}\cdot \mathbf{R}$ is an integer multiple of $2\pi$. It describes the spatial periodicities that can produce coherent Bragg scattering.

For direct primitive vectors ${\mathbf{a}}_i$ and oriented cell volume $\Omega ={\mathbf{a}}_1\cdot ({\mathbf{a}}_2\times {\mathbf{a}}_3)$, the primitive reciprocal vectors areThey obey ${\mathbf{a}}_i\cdot {\mathbf{b}}_j=2\pi {\delta }_{ij}$, and every reciprocal vector is an integer combination of them. Their units are inverse length. The reciprocal primitive cell volume is $(2\pi {)}^3/|\Omega |$, so expanding a direct lattice contracts its reciprocal scale.