What will be the packing fraction in the following case?
The sphere is touching the sides of the cube exactly.
We know that,
Packing fraction = Volume occupied inside the unit cellVolume of unit cell
Here the unit cell is the cube and volume occupied inside the unit cell will be equal to the volume of the sphere.
Let's assume the sphere is of radius r and cube is of side a.
Volume of sphere will be 43 ∗ pi ∗ r3
Volume of cube will be a3
Notice that the side of the cube is equal to the diameter of the sphere since it is touching all the sides.
So, a = d
a = 2r
Using the relation, we get
Packing fraction = 43 ∗ pi ∗ r3(2r)3
= 43 ∗ pi ∗ r38r3
= pi6 = 0.523