\ddagger X-ray scattering II
$\mathrm{BaTiO}_{3}$ has a primitive cubic lattice and a basis with atoms having fractional coordinates
Ba $[0,0,0]$
Ti $\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right]$
$\mathrm{O} \quad\left[\frac{1}{2}, \frac{1}{2}, 0\right], \quad\left[\frac{1}{2}, 0, \frac{1}{2}\right], \quad\left[0, \frac{1}{2}, \frac{1}{2}\right]$
P. Sketch the unit cell.
D. Show that the X-ray structure factor for the $(00 l)$ Bragg reflections is given by
$$
S_{(h k l)}=f_{B a}+(-1)^{l} f_{T i}+\left[1+2(-1)^{l}\right] f_{O}
$$
where $f_{\mathrm{Ba}}$ is the atomic form factor for $\mathrm{Ba}$, etc.
D Calculate the ratio $I_{(002)} / I_{(001)}$, where $I_{(h k l)}$ is the intensity of the X-ray diffraction from the ( $h k l$ ) planes. You may assume that the atomic form factor is proportional to atomic number $(Z)$, and neglect its dependence on the scattering vector. $\left(Z_{\mathrm{Ba}}=56, \quad Z_{\mathrm{Ti}}=22, \quad Z_{\mathrm{O}}=8 .\right)$