Rutile, TiO$2$ has a tetragonal crystal structure with a $=$ b $=0\mathrm{.}458$Rutile, $Ti{O}_2$ has a tetragonal crystal structure with $a=b=0\mathrm{.}458nm,c=0\mathrm{.}295nm$ and $==$

$=90.$ The atomic coordinates are : $(20$ points$)$

Ti: $000,\frac{1}{2}\frac{1}{2}\frac{1}{2}$

: u u $0,-u-u0,\frac{1}{2}+u,\frac{1}{2}-u\frac{1}{2},\frac{1}{2}-u\frac{1}{2}+u\frac{1}{2},$ where $u=0\mathrm{.}31$

a$)$ Draw the unit cell and indicate the $Ti$ and $O$ atoms in the unit cell. Draw an accurate

projection of the plane containing the $x$ and $y$ axes $($projection on $z$ plane$).$

b$)$ Determine the number of formula units per unit cell.

c$)$ Find the number of oxygen atoms surrounding each titanium atom and the approximate

shape of the coordination polyhedron.

d$)$ Derive an expression for the $Ti-O$ interatomic distances $($e$.$g$.$ considering the $Ti$ atom at $\frac{1}{2}$

$\frac{1}{21}\frac{?}{2}$ and calculate the actual distances. nm