Question

The unit cells can be represented by tennis balls arranged in various configurations in a box. For the FCC structure, (a) determine the lattice constant a in terms of atomic radius R, and (b) show the atomic packing factor (APF) is 0.74.

Answer 1

Based on the diagram shown in the attached file

R = Radius of the sphere  

a = Lattice constant

Using pythagoras theorem for the right angled triangle in figure 2

(4R)^2 = a^2 + a^2

16(R^2) = 2(a)^2

Solving the above equation

a=2R√2

To prove that the Atomic Packing factor is 0.74

Let the volume of atom in a unit cell be Vs

Total unit cell volume, Vc  

Vc = a^3

Vc =(2R√2)^3

Vc=16R^3 √2  =22.63R^3

Since there are 4 atoms in the FCC unit cell

Vs=4*(4/3 πR^3)

Vs=5.33 πR^3

APF=(5.33 πR^3)/(22.63R^3 )

APF=0.74