$Th{H}_2,$ an important nuclear reactor material, It has lattice parameters $a=b=4\mathrm{.}1,c=5\mathrm{.}03,===90.$ Th is at $(0,0,0),$ and $H$ is at $(0,\frac{1}{2},0)$ and $(\frac{1}{2},0,0).$ For each of these basis positions, I$-$centering is also active, i$.$e$.,$ must add the body centering operation to above positions. Label all the atoms in the unit cell.

$($a$)$ What is the Bravais lattice?

$($b$)$ List all the basis positions and how many formula units$/$unit cell?

$($c$)$ Draw the $3-$D unit cell and label all the atoms and lattice parameters. Also, present two diagrams, roughly to scale, which show the positions of all the atoms in the unit cell in projections down the $c$ and $b$ axes. Use the standard orientation for the axes. Mark the $z$ and $y$ coordinates of all the atoms on the diagrams. Finally, give the coordination number for the two atoms $($you can determine this by examining your unit cell, i$.$e$.,$ no need to calculate a radius ratio$).$

$($d$)$ What is the $3-$D point group?

$($e$)$ Draw below the diagrams for the point group: on left the symmetry element$($s$)$ and on right the coordinate$($s$)/$position$($s$)$ for the equipoint$($s$)$ For equipoints, use the notation from class $18/$slide $6$ on the bottom left.

Assume it is a single crystal:

$($f$)$ How many directions must the elastic constants be measured to fully describe the elastic modulus of $Th{H}_2?$

$($g$)$ How many directions must the thermal expansion coefficients be measured to fully describe the thermal expansion of $Th{H}_2?$

$($h$)$ How many directions must the piezoelectric constants be measured to fully describe the piezoelectricity of $Th{H}_2?$