# One-dimensional reciprocal lattice

I've seen that, a possible defining condition for the reciprocal lattice is:

$$\vec R_s \cdot \vec G=2 \pi l$$, where, $$R_s= n_1 \vec a_1+n_2 \vec a_2+n_3 \vec a_3$$ is the direct lattice, $$\vec G$$ is the reciprocal lattice, and $$l \in \mathbb{Z}$$

Using this definition in the one dimensional case, i find a wrong result, indeed: let $$a$$ be the distance between atoms, it follows that $$R_s=na$$ is the direct lattice. The defining equation, for the reciprocal lattice, is $$R_sG=2 \pi l$$. Substituting it becomes $$naG=2 \pi l$$ and so $$G=\frac {2 \pi l}{a n}$$, which is wrong, because it should be $$G=\frac {2 \pi l}{a }$$. Why do i have that extra integer, $$n$$, at the denominator?

In the formula $$\vec R_s \cdot \vec G = 2 \pi l$$, $$\vec R_s$$ are the basis vectors of the real lattice, rather than the entire lattice itself. You can work in the basis $$\vec R_s = a \hat x$$, in which case $$G = 2 \pi l/a$$, or you can choose a larger basis, like $$\vec R_s = 2 a \hat x$$, then $$G = 2 \pi l/2a$$. In this case, the output $$\vec G$$ is just the basis vector of the reciprocal space. It will of course be different if your real space lattice basis vector is different.
• $R_s$ is not necessarily the basis vectors - see here, for example. But $l$ is not an arbitrary integer that one is free to choose - rather it is a constraint that the product could be only integer. Feb 1 at 12:38