Question

You determine that material (c) has a hexagonal close-packed structure, where the c-to-a ratio is 1.633. Calculate the APF. Note: the center layer consists of the equivalent of three total atoms within the unit cell, and the radius of the atom is r.

Answer 1

Given: c-to-a ratio= 1.633 or c = 1.633a or a=c/1.633

(where c is the Height and a is the Length of each side)

Atomic Packing Factor (APF) is calculate as

APF= Volume of Atoms in Unit Cell/ Unit Cell Volume

In a hexagonal close-packed structure, number of atoms per unit cell is 6.

Thus, Volume of atoms in unit cell = 6*(4/3 $\pi$ r³)= 8 $\pi$ r³

where r is the Radius of the Atom

Unit Cell Volume= Area of hexagon* Height

Area of hexagon, A is

$A=\frac{3\sqrt{3}}{2}a^{2}$

Substitute for a,

$A=\frac{3\sqrt{3}}{2}(\frac{c}{1.633})^{2}=0.9742c^{2}$

Unit Cell Volume=0.9742c²*c

Unit Cell Volume= 0.9742 c³

Substitute the values to find APF

$APF=\frac{8\pi r^{3}}{0.9742c^{3}}$

In a hexagon, a=2r   $\Rightarrow$ r=a/2 or r= c/(1.633*2)=c/3.266

Thus the above equation becomes

$APF=\frac{8\pi (\frac{c}{3.266})^{3}}{0.9742c^{3}}= \frac{8\pi*c^{3}}{0.9742*c^{3}*3.266^{3}}$

$\boldsymbol{APF=0.7405}$