# Reciprocal lattice vectors

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?

• Well, those ‘other’ reciprocal space solutions would need to Fourier transform back to real space solutions. And that doesn’t happen for the Fourier transform. Apr 15, 2020 at 0:35

## 1 Answer

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 $$\mathbf{G} = m_i\mathbf{g}^i,\;m_i\in \mathbb{Z}$$