Sodium chloride has a face-centred cubic structure which can be regarded as a primitive cubic structure with a basis of sodium atoms at fractional coordinatese (0,0,0) (0.5,0.5,0) (0,0.5,0.5) and (0.5,0,0.5) and chlorine atoms at (0.5,0,0) (0,0.5,0) (0,0,0.5) and (0.5,0.5,0.5). The sodium ion has 10 electrons and the chlorine ion 18. Calculate the three different values of this structure factor. Show that the non zero values satify the relationship h+k=2n and k+l=2n where n is an integer.
I know that the structure factor is given by S=Sigma[bjexp(iK.r)] and that Shkl=Sbasis * Sffc
The Attempt at a Solution
Sbasis = N(sodium) +N(chloride)exp(-i*PI*h)
=10+18 = 28 if h, k l are all even
=10-18 = -8 if h,k, l are all odd
I know this is wrong because the answers should be 112, -36 and 0
But my notes also say that If h,k,l are all odd the Shkl=4N and if they are all even Shkl=4N and if one or more is odd and the others are even Shkl=0. I can see that you get the required numbers by using these three cases with the numbers calculated above but I don't know how it all fits together and you can justify multiplying them by four.
Also, I have no Idea how to show the relationships h+k=2n and k+l=2n