# Structure Factor for a Simple BCC Lattice

This is an example of a general misunderstanding I am having. The structure factor is given by $$S=\sum_{j}f_je^{i\mathbf{G}.\mathbf{x_j}}$$ where the index $j$ denotes a sum over the atoms within a unit cell. Suppose I have a BCC lattice made up of a single atom basis. My understanding is that I can choose any unit cell I like - so I will choose the Wigner-Seitz unit cell. In this case I have an atom at $[0,0,0]$ in each unit cell and my structure factor becomes $S=f_a$.

The method I am seeing everywhere else it to write the BCC lattice as a simple cublic lattice with a basis (i.e $[0,0,0]$ and $[\frac{1}2,\frac{1}2,\frac{1}2]$) and then you find $S=f_a[1+(-1)^{h+k+l}]$. Why is my approach incorrect?

The Wigner-Seitz cell of a bcc lattice is not cubic, i.e. the angles $\alpha$, $\beta$, $\gamma$ are not 90°. They are actually around 109.3°. Therefore you have to build your reciprocal lattice with this primitive cell and index the points by $h'k'l'$. As we know, the reciprocal lattice of a bcc lattice is fcc - so imagine the fcc Wigner-Seitz cell and use it for indexing.