In this problem, we will derive the structure factor for a sodium chloride-type unit cell. First, show that the coordinates of the cations at the eight corners are $(0,0,0),(1,0,0),(0,1,0)$, $(0,0,1),(1,1,0),(1,0,1),(0,1,1,)$, and $(1,1,1)$ and those at the six faces are $\left(\frac{1}{2}, \frac{1}{2}, 0\right),\left(\frac{1}{2}, 0, \frac{1}{2}\right)$ $\left(0, \frac{1}{2}, \frac{1}{2}\right),\left(\frac{1}{2}, \frac{1}{2}, 1\right),\left(\frac{1}{2}, 1, \frac{1}{2}\right)$, and $\left(1, \frac{1}{2}, \frac{1}{2}\right)$. Similarly, show that the coordinates of the anions along the 12 edges are $\left(\frac{1}{2}, 0,0\right),\left(0, \frac{1}{2}, 0\right),\left(0,0, \frac{1}{2}\right),\left(\frac{1}{2}, 1,0\right),\left(1, \frac{1}{2}, 0\right),\left(0, \frac{1}{2}, 1\right),\left(\frac{1}{2}, 0,1\right),\left(1,0, \frac{1}{2}\right)$, $\left(0,1, \frac{1}{2}\right),\left(\frac{1}{2}, 1,1\right),\left(1, \frac{1}{2}, 1\right)$, and $\left(1,1, \frac{1}{2}\right)$ and those of the anion at the center of the unit cell are $\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)$. Now show that
$$
\begin{aligned}
F(h k l)=& \frac{f_{+}}{8}\left[1+e^{2 \pi i h}+e^{2 \pi i k}+e^{2 \pi i l}+e^{2 \pi i(h+k)}+e^{2 \pi i(h+l)}+e^{2 \pi i(k+l)}+e^{2 \pi i(h+k+l)}\right] \\
&+\frac{f_{+}}{2}\left[e^{\pi i(h+k)}+e^{\pi i(h+l)}+e^{\pi i(k+l)}+e^{\pi i(h+k+2)}+e^{\pi i(h+2 k+l)}+e^{\pi i(2 h+k+l)}\right] \\
&+\frac{f_{-}}{4}\left[e^{\pi i h}+e^{\pi i k}+e^{\pi i l}+e^{\pi i(h+2 k)}+e^{\pi i(2 h+k)}+e^{\pi i(k+2 l)}+e^{\pi i(h+2 l)}+e^{\pi i(2 h+l)}+e^{\pi i(2 k+l)}\right.
\end{aligned}
$$
$$
\begin{aligned}
&\left.\quad+e^{\pi i(h+2 k+2 l)}+e^{\pi i(2 h+k+2 l)}+e^{n i(2 h+2 k+l)}\right]+f_{-} e^{\pi i(h+k+l)} \\
&=f_{+}\left[1+(-1)^{h+k}+(-1)^{h+l}+(-1)^{k+l}\right] \\
&\quad+f_{-}\left[(-1)^{h}+(-1)^{k}+(-1)^{l}+(-1)^{h+k+l}\right]
\end{aligned}
$$
Finally, show that
$$
F(h k l)=4\left(f_{+}+f_{-}\right)
$$
if $h, k$, and $l$ are all even; that
$$
F(h k l)=4\left(f_{+}-f_{-}\right)
$$
if $h, k$, and $l$ are all odd, and that $F(h k l)=0$ otherwise.