Problem $2$

Figure $($i$)$ shows a hard$-$sphere model for a hexagonal close$-$packed $($HCP$)$ unit cell, whereas Figures $($ii$)$

and $($iii$)$ are different representations to help you visualize the cell. The small spheres in Figure $($ii$)$ locate

the center of the atoms $($lattice points$)$ forming the unit cell. The parameter $a$ is the length of a hexagon

side, and $c$ is the height of the unit cell.

Based on the hard sphere$-$model of Figure $($i$),$ where the spheres are tangent to each other and atoms have

a radius $R,$ determine:

a$)$ Number of atoms per unit cell $(n).$

b$)$ The lattice parameters a and $c$ in terms of the radius $R$ of the atoms.

c$)$ Unit cell volume $V_c$ in terms of atomic radius $R.$

d$)$ Atomic Packing Factor $($APF$).$ How does it compare to the APF of FCC and BCC$?$