Reciprocal lattice vectors

Question

Given a lattice with primitive vectors $\{\mathbf{g_i}\}$, one can write the position of any lattice point as $\mathbf{R}_n = n^i \mathbf{g_i}$ with $n^i \in Z$. Quantites, such as electron density, have to be periodic on the lattice, that is, invariant to $\mathbf{R}_n$ translations. This condition implies that in the Fourier transform of those quantities only terms with wave vector $\mathbf{G}$ can appear for which

$\mathbf{G} \cdot \mathbf{R}_n = N$ with $N \in Z$

One can then introduce the reciprocal lattice vectors $\{\mathbf{g^i}\}$ and see that $\mathbf{G} = m_i \mathbf{g^i}$ with $m_i \in Z$ indeed satisfies the above equation. My question is, how can we see that these are the only solutions to the above equation? Is it possible that other $\mathbf{G}$ which cannot be written in the form $ m_i \mathbf{g^i}$ is also a solution? If not, why?

Answer 1

Your reciprocal lattice vectors should satisfy $$ \mathbf{G}\cdot\mathbf{R}_n\in \mathbb{Z}\quad \forall \mathbf{R}_n $$

in particular this includes your primative lattice vectors $\mathbf{g}_i$. If $\mathbf{G} = \mathbf{G}_0 + c \mathbf{g}^i$ with $\mathbf{G}_0\cdot\mathbf{g}_i = 0$ then $$ \mathbf{G}\cdot \mathbf{g}_i = c \Rightarrow c \in \mathbb{Z} $$ We can then apply the same argument to all $\mathbf{g}_i$ to conclude that $$ \mathbf{G} = m_i\mathbf{g}^i,\;m_i\in \mathbb{Z} $$