$(4)1.$ Niobium has an atomic radius of $0\mathrm{.}1430nm$ and a density of $8\mathrm{.}57\frac{g}{c}{m}^3.$

$($Avogadro$\text{'}$s number $N_A=6\mathrm{.}022{10}^{23}$ atoms $\frac{?}{\text{m}\text{o}\text{l}}$ and atomic weight of $Nb,A_{Nb}=92\mathrm{.}91$

$\frac{g}{m}ol)$

$($a$)$ Determine whether it has an FCC or BCC crystal structure?

$($b$)$ For which set of crystallographic planes will a first$-$order diffraction peak occur at a

diffraction angle $(2)$ of $75\mathrm{.}99$ when monochromatic $x-$ray radiation having a wavelength

of $0\mathrm{.}1659nm$ is used?

$($c$)$ Sketch one of the crystallographic planes of this set within a cubic unit cell that you

have found in part $($b$).$

$($d$)$ Calculate the planar density of the plane you have drawn in part $($c$)$ in atoms $\frac{?}{c{m}^2}.$

$(4)2.$ A substance is known to crystallize into a base$-$centered tetragonal cell as shown in

the figure. In this cell, $a=1\mathrm{.}2{10}^{-8}cm$ and $c=24x{10}^{-8}cm$

$($a$)$ How many atoms are there per unit cell?

$($b$)$ Assuming that the atoms are in close contact along the diagonal directions on the

base, calculate the atomic packing factor.

$($c$)$ Calculate the planar density of the most densely packed plane of this crystal structure

in atom $\frac{?}{c{m}^2}.$ Index and sketch the atomic arrangement of this plane.

$($d$)$ Calculate the theoretical density of this substance assuming that the atomic weight is

$12$gram$\frac{s}{m}ol$ and $N_A=6\mathrm{.}022{10}^{23}$ atoms $\frac{?}{\text{m}\text{o}\text{l}}.$

$(2)3.$ Answer the following questions based on the following table.

$($a$)$ Determine the coordination number and the coordination geometry of ZnO.

$($b$)$ Calculate the ionic packing factor for ZnO where anions are in a cubic close packed

arrangement and the cations are in all of the octahedral holes. Also draw the unit cell

structure.