Fourier Expansion of the Electron Density

Since the lattice has a periodicity with a repeat distance a in one dimension, we would expect the electron density surrounding the atoms in any unit cell to have the same periodicity. Thus we should be able to represent the electron density of the solid by a Fourier series of the form 

The tilde on Fourier coefficients is to remind us that they are complex. 

The electron density should be invariant under any translation operation T such that n x + T) = n x), where T = wxa. In other words, the electron density in one cell should look the same as in any other cell. We can see that this is the case from 

---