Maybe a bit late, but I have a question about the structure factor of the FCC lattice.
The answer model (and the lecture notes for X-rays) state that the structure factor of the FCC latice is S = f × (1 + e^{iπ(h+l)} + e^{iπ(h+k)} + e^{iπ(k+l)}). I believe the reasoning is that (for example) for the second term we have r=(a_1+a_2)/2 and G=hb_1+kb_2+lb_3, thus G\dot r=(ha_1*b_1+ka_2b_2)/2 and then a_i*b_i=2\pi is used. My problem is: earlier in the lecture notes, it was stated that the last equation was true for the primitive lattice vectors, but the a_i that are used in the problem are conventional lattice vectors. When I calculate the primitive vectors, I get G=2\pi/a*((-h+k+l)x+(h-k+l)y+(h+k-l)z) and the structure factor becomes f(1+e^{2\pi h}+e^{2\pi k}+e^{2\pi l})=4f.
I was wondering what part of my reasoning is wrong.