Obtain an expression for the structure factor for the following unit cells and explain the absence of some reflections in each of these systems shown in the table given below.

a$.$ A face$-$centered cubic cell with four atoms of the same kind per unit cell. $($Assume that they are located at $000,1\frac{1}{2}\frac{1}{2}0,1\frac{1}{2}0\frac{1}{2}$ and $0\frac{1}{2}\frac{1}{2}).$

b$.$ A body$-$centered cubic cell with two atoms of the same kind per unit cell. $($Assume that they are located at $000$ and $1\frac{1}{2}\frac{1}{2}\frac{1}{2}\frac{?}{2}.$

Recall:

The structure factor $F$ is given by:

$F=f_1{e}^{2i(h{u}_1+k{v}_1+l{w}_1)}+f_2{e}^{2i(h{u}_2+k{v}_2+l{w}_2)}+f_3{e}^{2i(h{u}_3+k{v}_3+l{w}_3)}+$cdots.

a$)$

$e^i=e^{3i}=e^{5i}=-1,$

b$)$

$e^{2i}=e^{4i}=e^{6i}=+1,$

c$)$

$in$ general, $,e^{ni}=(-1{)}^n,$

where $n$ is any integer,

d$)$

$e^{ni}=e^{-ni},$ where $n$ is any integer,

e$)$

$e^{ix}+e^{-ix}=2cosx.$

Table $2\mathrm{.}1$ Allowed and forbidden reflections in crystals of cubic symmetry

$\backslash$table$[[h^2+k^2+l^2,$Primitive cubic,Face$-$centred cubic,Body$-$centred cubic$],[1,100,-,-],[2,110,-,110],[3,111,111,200],[4,200,200,-],[5,210,-,211],[6,211,-,-],[7,-,-,220],[8,220,220,-],[9,\frac{221}{300},-,310],[10,310,-,-],[11,311,311,222],[12,222,-,-],[13,320,-,321],[14,321,-,-],[15,-,400,400],[16,400,222,]]$