# If my lattice has an atomic basis, do I also find the reciprocals of the basis vectors to get the reciprocal crystal structure?

That is what my crystal structure looks like. The blue atoms sit on every lattice point (basis vector of $\{0,0\}$) and the red atoms have basis vector of $\left\{{2\over3},{1\over3}\right\}$. The lattice vectors are:

$$a~=~\left\{-{1\over2}, -{\sqrt3\over2}\right\},$$$$b~=~\{1, 0\}.$$

I used these two lattice vectors and found their reciprocal ones, and plotted them:

I know that the blue atoms are correct because I know from previous experience that the reciprocal lattice of the hexagonal structure just looks like the direct lattice but rotated $45^\circ$. The problem is the location of the red atoms. To plot the above I took the basis vectors and displaced them by the reciprocal vectors. I don't know if that's correct.

The other alternative is to find the reciprocals of the basis vectors too and then plot them. Which is here:

So do I have to find the reciprocals of the basis vectors or not? The literature only specifies finding the reciprocals of the lattice vectors, but no mention of what to do when there's an atomic basis.

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In a way, the reciprocal lattice is the Fourier transform of the original lattice. Now it's in the nature of the Fourier transform to change a sub- into a superstructure.

That means that the basis vectors (i.e. the sub-structure of your unit cell) in real space lead to a super-structure in reciprocal space.

So, the reciprocal lattice vectors define a reciprocal lattice, and the extra atoms in the basis lead to certain selection rules for which of the reciprocal lattice vectors you'd be able to find in, e.g., x-ray scattering, because they determine the structure factor of your particular crystal lattice.

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